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Kaisa Matomaki, Maksym Radziwill, and I have uploaded to the arXiv our paper “Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges“, submitted to Proceedings of the London Mathematical Society. This paper is concerned with the estimation of correlations such as

\displaystyle \sum_{n \leq X} \Lambda(n) \Lambda(n+h) \ \ \ \ \ (1)

for medium-sized {h} and large {X}, where {\Lambda} is the von Mangoldt function; we also consider variants of this sum in which one of the von Mangoldt functions is replaced with a (higher order) divisor function, but for sake of discussion let us focus just on the sum (1). Understanding this sum is very closely related to the problem of finding pairs of primes that differ by {h}; for instance, if one could establish a lower bound

\displaystyle \sum_{n \leq X} \Lambda(n) \Lambda(n+2) \gg X

then this would easily imply the twin prime conjecture.

The (first) Hardy-Littlewood conjecture asserts an asymptotic

\displaystyle \sum_{n \leq X} \Lambda(n) \Lambda(n+h) = {\mathfrak S}(h) X + o(X) \ \ \ \ \ (2)

as {X \rightarrow \infty} for any fixed positive {h}, where the singular series {{\mathfrak S}(h)} is an arithmetic factor arising from the irregularity of distribution of {\Lambda} at small moduli, defined explicitly by

\displaystyle {\mathfrak S}(h) := 2 \Pi_2 \prod_{p|h; p>2} \frac{p-2}{p-1}

when {h} is even, and {{\mathfrak S}(h)=0} when {h} is odd, where

\displaystyle \Pi_2 := \prod_{p>2} (1-\frac{1}{(p-1)^2}) = 0.66016\dots

is (half of) the twin prime constant. See for instance this previous blog post for a a heuristic explanation of this conjecture. From the previous discussion we see that (2) for {h=2} would imply the twin prime conjecture. Sieve theoretic methods are only able to provide an upper bound of the form { \sum_{n \leq X} \Lambda(n) \Lambda(n+h) \ll {\mathfrak S}(h) X}.

Needless to say, apart from the trivial case of odd {h}, there are no values of {h} for which the Hardy-Littlewood conjecture is known. However there are some results that say that this conjecture holds “on the average”: in particular, if {H} is a quantity depending on {X} that is somewhat large, there are results that show that (2) holds for most (i.e. for {1-o(1)}) of the {h} betwen {0} and {H}. Ideally one would like to get {H} as small as possible, in particular one can view the full Hardy-Littlewood conjecture as the endpoint case when {H} is bounded.

The first results in this direction were by van der Corput and by Lavrik, who established such a result with {H = X} (with a subsequent refinement by Balog); Wolke lowered {H} to {X^{5/8+\varepsilon}}, and Mikawa lowered {H} further to {X^{1/3+\varepsilon}}. The main result of this paper is a further lowering of {H} to {X^{8/33+\varepsilon}}. In fact (as in the preceding works) we get a better error term than {o(X)}, namely an error of the shape {O_A( X \log^{-A} X)} for any {A}.

Our arguments initially proceed along standard lines. One can use the Hardy-Littlewood circle method to express the correlation in (2) as an integral involving exponential sums {S(\alpha) := \sum_{n \leq X} \Lambda(n) e(\alpha n)}. The contribution of “major arc” {\alpha} is known by a standard computation to recover the main term {{\mathfrak S}(h) X} plus acceptable errors, so it is a matter of controlling the “minor arcs”. After averaging in {h} and using the Plancherel identity, one is basically faced with establishing a bound of the form

\displaystyle \int_{\beta-1/H}^{\beta+1/H} |S(\alpha)|^2\ d\alpha \ll_A X \log^{-A} X

for any “minor arc” {\beta}. If {\beta} is somewhat close to a low height rational {a/q} (specifically, if it is within {X^{-1/6-\varepsilon}} of such a rational with {q = O(\log^{O(1)} X)}), then this type of estimate is roughly of comparable strength (by another application of Plancherel) to the best available prime number theorem in short intervals on the average, namely that the prime number theorem holds for most intervals of the form {[x, x + x^{1/6+\varepsilon}]}, and we can handle this case using standard mean value theorems for Dirichlet series. So we can restrict attention to the “strongly minor arc” case where {\beta} is far from such rationals.

The next step (following some ideas we found in a paper of Zhan) is to rewrite this estimate not in terms of the exponential sums {S(\alpha) := \sum_{n \leq X} \Lambda(n) e(\alpha n)}, but rather in terms of the Dirichlet polynomial {F(s) := \sum_{n \sim X} \frac{\Lambda(n)}{n^s}}. After a certain amount of computation (including some oscillatory integral estimates arising from stationary phase), one is eventually reduced to the task of establishing an estimate of the form

\displaystyle \int_{t \sim \lambda X} (\sum_{t-\lambda H}^{t+\lambda H} |F(\frac{1}{2}+it')|\ dt')^2\ dt \ll_A \lambda^2 H^2 X \log^{-A} X

for any {X^{-1/6-\varepsilon} \ll \lambda \ll \log^{-B} X} (with {B} sufficiently large depending on {A}).

The next step, which is again standard, is the use of the Heath-Brown identity (as discussed for instance in this previous blog post) to split up {\Lambda} into a number of components that have a Dirichlet convolution structure. Because the exponent {8/33} we are shooting for is less than {1/4}, we end up with five types of components that arise, which we call “Type {d_1}“, “Type {d_2}“, “Type {d_3}“, “Type {d_4}“, and “Type II”. The “Type II” sums are Dirichlet convolutions involving a factor supported on a range {[X^\varepsilon, X^{-\varepsilon} H]} and is quite easy to deal with; the “Type {d_j}” terms are Dirichlet convolutions that resemble (non-degenerate portions of) the {j^{th}} divisor function, formed from convolving together {j} portions of {1}. The “Type {d_1}” and “Type {d_2}” terms can be estimated satisfactorily by standard moment estimates for Dirichlet polynomials; this already recovers the result of Mikawa (and our argument is in fact slightly more elementary in that no Kloosterman sum estimates are required). It is the treatment of the “Type {d_3}” and “Type {d_4}” sums that require some new analysis, with the Type {d_3} terms turning to be the most delicate. After using an existing moment estimate of Jutila for Dirichlet L-functions, matters reduce to obtaining a family of estimates, a typical one of which (relating to the more difficult Type {d_3} sums) is of the form

\displaystyle \int_{t - H}^{t+H} |M( \frac{1}{2} + it')|^2\ dt' \ll X^{\varepsilon^2} H \ \ \ \ \ (3)

for “typical” ordinates {t} of size {X}, where {M} is the Dirichlet polynomial {M(s) := \sum_{n \sim X^{1/3}} \frac{1}{n^s}} (a fragment of the Riemann zeta function). The precise definition of “typical” is a little technical (because of the complicated nature of Jutila’s estimate) and will not be detailed here. Such a claim would follow easily from the Lindelof hypothesis (which would imply that {M(1/2 + it) \ll X^{o(1)}}) but of course we would like to have an unconditional result.

At this point, having exhausted all the Dirichlet polynomial estimates that are usefully available, we return to “physical space”. Using some further Fourier-analytic and oscillatory integral computations, we can estimate the left-hand side of (3) by an expression that is roughly of the shape

\displaystyle \frac{H}{X^{1/3}} \sum_{\ell \sim X^{1/3}/H} |\sum_{m \sim X^{1/3}} e( \frac{t}{2\pi} \log \frac{m+\ell}{m-\ell} )|.

The phase {\frac{t}{2\pi} \log \frac{m+\ell}{m-\ell}} can be Taylor expanded as the sum of {\frac{t_j \ell}{\pi m}} and a lower order term {\frac{t_j \ell^3}{3\pi m^3}}, plus negligible errors. If we could discard the lower order term then we would get quite a good bound using the exponential sum estimates of Robert and Sargos, which control averages of exponential sums with purely monomial phases, with the averaging allowing us to exploit the hypothesis that {t} is “typical”. Figuring out how to get rid of this lower order term caused some inefficiency in our arguments; the best we could do (after much experimentation) was to use Fourier analysis to shorten the sums, estimate a one-parameter average exponential sum with a binomial phase by a two-parameter average with a monomial phase, and then use the van der Corput {B} process followed by the estimates of Robert and Sargos. This rather complicated procedure works up to {H = X^{8/33+\varepsilon}} it may be possible that some alternate way to proceed here could improve the exponent somewhat.

In a sequel to this paper, we will use a somewhat different method to reduce {H} to a much smaller value of {\log^{O(1)} X}, but only if we replace the correlations {\sum_{n \leq X} \Lambda(n) \Lambda(n+h)} by either {\sum_{n \leq X} \Lambda(n) d_k(n+h)} or {\sum_{n \leq X} d_k(n) d_l(n+h)}, and also we now only save a {o(1)} in the error term rather than {O_A(\log^{-A} X)}.

Given a function {f: {\bf N} \rightarrow \{-1,+1\}} on the natural numbers taking values in {+1, -1}, one can invoke the Furstenberg correspondence principle to locate a measure preserving system {T \circlearrowright (X, \mu)} – a probability space {(X,\mu)} together with a measure-preserving shift {T: X \rightarrow X} (or equivalently, a measure-preserving {{\bf Z}}-action on {(X,\mu)}) – together with a measurable function (or “observable”) {F: X \rightarrow \{-1,+1\}} that has essentially the same statistics as {f} in the sense that

\displaystyle \lim \inf_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N f(n+h_1) \dots f(n+h_k)

\displaystyle \leq \int_X F(T^{h_1} x) \dots F(T^{h_k} x)\ d\mu(x)

\displaystyle \leq \lim \sup_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N f(n+h_1) \dots f(n+h_k)

for any integers {h_1,\dots,h_k}. In particular, one has

\displaystyle \int_X F(T^{h_1} x) \dots F(T^{h_k} x)\ d\mu(x) = \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N f(n+h_1) \dots f(n+h_k) \ \ \ \ \ (1)

 

whenever the limit on the right-hand side exists. We will refer to the system {T \circlearrowright (X,\mu)} together with the designated function {F} as a Furstenberg limit ot the sequence {f}. These Furstenberg limits capture some, but not all, of the asymptotic behaviour of {f}; roughly speaking, they control the typical “local” behaviour of {f}, involving correlations such as {\frac{1}{N} \sum_{n=1}^N f(n+h_1) \dots f(n+h_k)} in the regime where {h_1,\dots,h_k} are much smaller than {N}. However, the control on error terms here is usually only qualitative at best, and one usually does not obtain non-trivial control on correlations in which the {h_1,\dots,h_k} are allowed to grow at some significant rate with {N} (e.g. like some power {N^\theta} of {N}).

The correspondence principle is discussed in these previous blog posts. One way to establish the principle is by introducing a Banach limit {p\!-\!\lim: \ell^\infty({\bf N}) \rightarrow {\bf R}} that extends the usual limit functional on the subspace of {\ell^\infty({\bf N})} consisting of convergent sequences while still having operator norm one. Such functionals cannot be constructed explicitly, but can be proven to exist (non-constructively and non-uniquely) using the Hahn-Banach theorem; one can also use a non-principal ultrafilter here if desired. One can then seek to construct a system {T \circlearrowright (X,\mu)} and a measurable function {F: X \rightarrow \{-1,+1\}} for which one has the statistics

\displaystyle \int_X F(T^{h_1} x) \dots F(T^{h_k} x)\ d\mu(x) = p\!-\!\lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N f(n+h_1) \dots f(n+h_k) \ \ \ \ \ (2)

 

for all {h_1,\dots,h_k}. One can explicitly construct such a system as follows. One can take {X} to be the Cantor space {\{-1,+1\}^{\bf Z}} with the product {\sigma}-algebra and the shift

\displaystyle T ( (x_n)_{n \in {\bf Z}} ) := (x_{n+1})_{n \in {\bf Z}}

with the function {F: X \rightarrow \{-1,+1\}} being the coordinate function at zero:

\displaystyle F( (x_n)_{n \in {\bf Z}} ) := x_0

(so in particular {F( T^h (x_n)_{n \in {\bf Z}} ) = x_h} for any {h \in {\bf Z}}). The only thing remaining is to construct the invariant measure {\mu}. In order to be consistent with (2), one must have

\displaystyle \mu( \{ (x_n)_{n \in {\bf Z}}: x_{h_j} = \epsilon_j \forall 1 \leq j \leq k \} )

\displaystyle = p\!-\!\lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N 1_{f(n+h_1)=\epsilon_1} \dots 1_{f(n+h_k)=\epsilon_k}

for any distinct integers {h_1,\dots,h_k} and signs {\epsilon_1,\dots,\epsilon_k}. One can check that this defines a premeasure on the Boolean algebra of {\{-1,+1\}^{\bf Z}} defined by cylinder sets, and the existence of {\mu} then follows from the Hahn-Kolmogorov extension theorem (or the closely related Kolmogorov extension theorem). One can then check that the correspondence (2) holds, and that {\mu} is translation-invariant; the latter comes from the translation invariance of the (Banach-)Césaro averaging operation {f \mapsto p\!-\!\lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N f(n)}. A variant of this construction shows that the Furstenberg limit is unique up to equivalence if and only if all the limits appearing in (1) actually exist.

One can obtain a slightly tighter correspondence by using a smoother average than the Césaro average. For instance, one can use the logarithmic Césaro averages {\lim_{N \rightarrow \infty} \frac{1}{\log N}\sum_{n=1}^N \frac{f(n)}{n}} in place of the Césaro average {\sum_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N f(n)}, thus one replaces (2) by

\displaystyle \int_X F(T^{h_1} x) \dots F(T^{h_k} x)\ d\mu(x)

\displaystyle = p\!-\!\lim_{N \rightarrow \infty} \frac{1}{\log N} \sum_{n=1}^N \frac{f(n+h_1) \dots f(n+h_k)}{n}.

Whenever the Césaro average of a bounded sequence {f: {\bf N} \rightarrow {\bf R}} exists, then the logarithmic Césaro average exists and is equal to the Césaro average. Thus, a Furstenberg limit constructed using logarithmic Banach-Césaro averaging still obeys (1) for all {h_1,\dots,h_k} when the right-hand side limit exists, but also obeys the more general assertion

\displaystyle \int_X F(T^{h_1} x) \dots F(T^{h_k} x)\ d\mu(x)

\displaystyle = \lim_{N \rightarrow \infty} \frac{1}{\log N} \sum_{n=1}^N \frac{f(n+h_1) \dots f(n+h_k)}{n}

whenever the limit of the right-hand side exists.

In a recent paper of Frantizinakis, the Furstenberg limits of the Liouville function {\lambda} (with logarithmic averaging) were studied. Some (but not all) of the known facts and conjectures about the Liouville function can be interpreted in the Furstenberg limit. For instance, in a recent breakthrough result of Matomaki and Radziwill (discussed previously here), it was shown that the Liouville function exhibited cancellation on short intervals in the sense that

\displaystyle \lim_{H \rightarrow \infty} \limsup_{X \rightarrow \infty} \frac{1}{X} \int_X^{2X} \frac{1}{H} |\sum_{x \leq n \leq x+H} \lambda(n)|\ dx = 0.

In terms of Furstenberg limits of the Liouville function, this assertion is equivalent to the assertion that

\displaystyle \lim_{H \rightarrow \infty} \int_X |\frac{1}{H} \sum_{h=1}^H F(T^h x)|\ d\mu(x) = 0

for all Furstenberg limits {T \circlearrowright (X,\mu), F} of Liouville (including those without logarithmic averaging). Invoking the mean ergodic theorem (discussed in this previous post), this assertion is in turn equivalent to the observable {F} that corresponds to the Liouville function being orthogonal to the invariant factor {L^\infty(X,\mu)^{\bf Z} = \{ g \in L^\infty(X,\mu): g \circ T = g \}} of {X}; equivalently, the first Gowers-Host-Kra seminorm {\|F\|_{U^1(X)}} of {F} (as defined for instance in this previous post) vanishes. The Chowla conjecture, which asserts that

\displaystyle \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N \lambda(n+h_1) \dots \lambda(n+h_k) = 0

for all distinct integers {h_1,\dots,h_k}, is equivalent to the assertion that all the Furstenberg limits of Liouville are equivalent to the Bernoulli system ({\{-1,+1\}^{\bf Z}} with the product measure arising from the uniform distribution on {\{-1,+1\}}, with the shift {T} and observable {F} as before). Similarly, the logarithmically averaged Chowla conjecture

\displaystyle \lim_{N \rightarrow \infty} \frac{1}{\log N} \sum_{n=1}^N \frac{\lambda(n+h_1) \dots \lambda(n+h_k)}{n} = 0

is equivalent to the assertion that all the Furstenberg limits of Liouville with logarithmic averaging are equivalent to the Bernoulli system. Recently, I was able to prove the two-point version

\displaystyle \lim_{N \rightarrow \infty} \frac{1}{\log N} \sum_{n=1}^N \frac{\lambda(n) \lambda(n+h)}{n} = 0 \ \ \ \ \ (3)

 

of the logarithmically averaged Chowla conjecture, for any non-zero integer {h}; this is equivalent to the perfect strong mixing property

\displaystyle \int_X F(x) F(T^h x)\ d\mu(x) = 0

for any Furstenberg limit of Liouville with logarithmic averaging, and any {h \neq 0}.

The situation is more delicate with regards to the Sarnak conjecture, which is equivalent to the assertion that

\displaystyle \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N \lambda(n) f(n) = 0

for any zero-entropy sequence {f: {\bf N} \rightarrow {\bf R}} (see this previous blog post for more discussion). Morally speaking, this conjecture should be equivalent to the assertion that any Furstenberg limit of Liouville is disjoint from any zero entropy system, but I was not able to formally establish an implication in either direction due to some technical issues regarding the fact that the Furstenberg limit does not directly control long-range correlations, only short-range ones. (There are however ergodic theoretic interpretations of the Sarnak conjecture that involve the notion of generic points; see this paper of El Abdalaoui, Lemancyk, and de la Rue.) But the situation is currently better with the logarithmically averaged Sarnak conjecture

\displaystyle \lim_{N \rightarrow \infty} \frac{1}{\log N} \sum_{n=1}^N \frac{\lambda(n) f(n)}{n} = 0,

as I was able to show that this conjecture was equivalent to the logarithmically averaged Chowla conjecture, and hence to all Furstenberg limits of Liouville with logarithmic averaging being Bernoulli; I also showed the conjecture was equivalent to local Gowers uniformity of the Liouville function, which is in turn equivalent to the function {F} having all Gowers-Host-Kra seminorms vanishing in every Furstenberg limit with logarithmic averaging. In this recent paper of Frantzikinakis, this analysis was taken further, showing that the logarithmically averaged Chowla and Sarnak conjectures were in fact equivalent to the much milder seeming assertion that all Furstenberg limits with logarithmic averaging were ergodic.

Actually, the logarithmically averaged Furstenberg limits have more structure than just a {{\bf Z}}-action on a measure preserving system {(X,\mu)} with a single observable {F}. Let {Aff_+({\bf Z})} denote the semigroup of affine maps {n \mapsto an+b} on the integers with {a,b \in {\bf Z}} and {a} positive. Also, let {\hat {\bf Z}} denote the profinite integers (the inverse limit of the cyclic groups {{\bf Z}/q{\bf Z}}). Observe that {Aff_+({\bf Z})} acts on {\hat {\bf Z}} by taking the inverse limit of the obvious actions of {Aff_+({\bf Z})} on {{\bf Z}/q{\bf Z}}.

Proposition 1 (Enriched logarithmically averaged Furstenberg limit of Liouville) Let {p\!-\!\lim} be a Banach limit. Then there exists a probability space {(X,\mu)} with an action {\phi \mapsto T^\phi} of the affine semigroup {Aff_+({\bf Z})}, as well as measurable functions {F: X \rightarrow \{-1,+1\}} and {M: X \rightarrow \hat {\bf Z}}, with the following properties:

  • (i) (Affine Furstenberg limit) For any {\phi_1,\dots,\phi_k \in Aff_+({\bf Z})}, and any congruence class {a\ (q)}, one has

    \displaystyle p\!-\!\lim_{N \rightarrow \infty} \frac{1}{\log N} \sum_{n=1}^N \frac{\lambda(\phi_1(n)) \dots \lambda(\phi_k(n)) 1_{n = a\ (q)}}{n}

    \displaystyle = \int_X F( T^{\phi_1}(x) ) \dots F( T^{\phi_k}(x) ) 1_{M(x) = a\ (q)}\ d\mu(x).

  • (ii) (Equivariance of {M}) For any {\phi \in Aff_+({\bf Z})}, one has

    \displaystyle M( T^\phi(x) ) = \phi( M(x) )

    for {\mu}-almost every {x \in X}.

  • (iii) (Multiplicativity at fixed primes) For any prime {p}, one has

    \displaystyle F( T^{p\cdot} x ) = - F(x)

    for {\mu}-almost every {x \in X}, where {p \cdot \in Aff_+({\bf Z})} is the dilation map {n \mapsto pn}.

  • (iv) (Measure pushforward) If {\phi \in Aff_+({\bf Z})} is of the form {\phi(n) = an+b} and {S_\phi \subset X} is the set {S_\phi = \{ x \in X: M(x) \in \phi(\hat {\bf Z}) \}}, then the pushforward {T^\phi_* \mu} of {\mu} by {\phi} is equal to {a \mu\downharpoonright_{S_\phi}}, that is to say one has

    \displaystyle \mu( (T^\phi)^{-1}(E) ) = a \mu( E \cap S_\phi )

    for every measurable {E \subset X}.

Note that {{\bf Z}} can be viewed as the subgroup of {Aff_+({\bf Z})} consisting of the translations {n \mapsto n + b}. If one only keeps the {{\bf Z}}-portion of the {Aff_+({\bf Z})} action and forgets the rest (as well as the function {M}) then the action becomes measure-preserving, and we recover an ordinary Furstenberg limit with logarithmic averaging. However, the additional structure here can be quite useful; for instance, one can transfer the proof of (3) to this setting, which we sketch below the fold, after proving the proposition.

The observable {M}, roughly speaking, means that points {x} in the Furstenberg limit {X} constructed by this proposition are still “virtual integers” in the sense that one can meaningfully compute the residue class of {x} modulo any natural number modulus {q}, by first applying {M} and then reducing mod {q}. The action of {Aff_+({\bf Z})} means that one can also meaningfully multiply {x} by any natural number, and translate it by any integer. As with other applications of the correspondence principle, the main advantage of moving to this more “virtual” setting is that one now acquires a probability measure {\mu}, so that the tools of ergodic theory can be readily applied.

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Given a random variable {X} that takes on only finitely many values, we can define its Shannon entropy by the formula

\displaystyle  H(X) := \sum_x \mathbf{P}(X=x) \log \frac{1}{\mathbf{P}(X=x)}

with the convention that {0 \log \frac{1}{0} = 0}. (In some texts, one uses the logarithm to base {2} rather than the natural logarithm, but the choice of base will not be relevant for this discussion.) This is clearly a nonnegative quantity. Given two random variables {X,Y} taking on finitely many values, the joint variable {(X,Y)} is also a random variable taking on finitely many values, and also has an entropy {H(X,Y)}. It obeys the Shannon inequalities

\displaystyle  H(X), H(Y) \leq H(X,Y) \leq H(X) + H(Y)

so we can define some further nonnegative quantities, the mutual information

\displaystyle  I(X:Y) := H(X) + H(Y) - H(X,Y)

and the conditional entropies

\displaystyle  H(X|Y) := H(X,Y) - H(Y); \quad H(Y|X) := H(X,Y) - H(X).

More generally, given three random variables {X,Y,Z}, one can define the conditional mutual information

\displaystyle  I(X:Y|Z) := H(X|Z) + H(Y|Z) - H(X,Y|Z)

and the final of the Shannon entropy inequalities asserts that this quantity is also non-negative.

The mutual information {I(X:Y)} is a measure of the extent to which {X} and {Y} fail to be independent; indeed, it is not difficult to show that {I(X:Y)} vanishes if and only if {X} and {Y} are independent. Similarly, {I(X:Y|Z)} vanishes if and only if {X} and {Y} are conditionally independent relative to {Z}. At the other extreme, {H(X|Y)} is a measure of the extent to which {X} fails to depend on {Y}; indeed, it is not difficult to show that {H(X|Y)=0} if and only if {X} is determined by {Y} in the sense that there is a deterministic function {f} such that {X = f(Y)}. In a related vein, if {X} and {X'} are equivalent in the sense that there are deterministic functional relationships {X = f(X')}, {X' = g(X)} between the two variables, then {X} is interchangeable with {X'} for the purposes of computing the above quantities, thus for instance {H(X) = H(X')}, {H(X,Y) = H(X',Y)}, {I(X:Y) = I(X':Y)}, {I(X:Y|Z) = I(X':Y|Z)}, etc..

One can get some initial intuition for these information-theoretic quantities by specialising to a simple situation in which all the random variables {X} being considered come from restricting a single random (and uniformly distributed) boolean function {F: \Omega \rightarrow \{0,1\}} on a given finite domain {\Omega} to some subset {A} of {\Omega}:

\displaystyle  X = F \downharpoonright_A.

In this case, {X} has the law of a random uniformly distributed boolean function from {A} to {\{0,1\}}, and the entropy here can be easily computed to be {|A| \log 2}, where {|A|} denotes the cardinality of {A}. If {X} is the restriction of {F} to {A}, and {Y} is the restriction of {F} to {B}, then the joint variable {(X,Y)} is equivalent to the restriction of {F} to {A \cup B}. If one discards the normalisation factor {\log 2}, one then obtains the following dictionary between entropy and the combinatorics of finite sets:

Random variables {X,Y,Z} Finite sets {A,B,C}
Entropy {H(X)} Cardinality {|A|}
Joint variable {(X,Y)} Union {A \cup B}
Mutual information {I(X:Y)} Intersection cardinality {|A \cap B|}
Conditional entropy {H(X|Y)} Set difference cardinality {|A \backslash B|}
Conditional mutual information {I(X:Y|Z)} {|(A \cap B) \backslash C|}
{X, Y} independent {A, B} disjoint
{X} determined by {Y} {A} a subset of {B}
{X,Y} conditionally independent relative to {Z} {A \cap B \subset C}

Every (linear) inequality or identity about entropy (and related quantities, such as mutual information) then specialises to a combinatorial inequality or identity about finite sets that is easily verified. For instance, the Shannon inequality {H(X,Y) \leq H(X)+H(Y)} becomes the union bound {|A \cup B| \leq |A| + |B|}, and the definition of mutual information becomes the inclusion-exclusion formula

\displaystyle  |A \cap B| = |A| + |B| - |A \cup B|.

For a more advanced example, consider the data processing inequality that asserts that if {X, Z} are conditionally independent relative to {Y}, then {I(X:Z) \leq I(X:Y)}. Specialising to sets, this now says that if {A, C} are disjoint outside of {B}, then {|A \cap C| \leq |A \cap B|}; this can be made apparent by considering the corresponding Venn diagram. This dictionary also suggests how to prove the data processing inequality using the existing Shannon inequalities. Firstly, if {A} and {C} are not necessarily disjoint outside of {B}, then a consideration of Venn diagrams gives the more general inequality

\displaystyle  |A \cap C| \leq |A \cap B| + |(A \cap C) \backslash B|

and a further inspection of the diagram then reveals the more precise identity

\displaystyle  |A \cap C| + |(A \cap B) \backslash C| = |A \cap B| + |(A \cap C) \backslash B|.

Using the dictionary in the reverse direction, one is then led to conjecture the identity

\displaystyle  I( X : Z ) + I( X : Y | Z ) = I( X : Y ) + I( X : Z | Y )

which (together with non-negativity of conditional mutual information) implies the data processing inequality, and this identity is in turn easily established from the definition of mutual information.

On the other hand, not every assertion about cardinalities of sets generalises to entropies of random variables that are not arising from restricting random boolean functions to sets. For instance, a basic property of sets is that disjointness from a given set {C} is preserved by unions:

\displaystyle  A \cap C = B \cap C = \emptyset \implies (A \cup B) \cap C = \emptyset.

Indeed, one has the union bound

\displaystyle  |(A \cup B) \cap C| \leq |A \cap C| + |B \cap C|. \ \ \ \ \ (1)

Applying the dictionary in the reverse direction, one might now conjecture that if {X} was independent of {Z} and {Y} was independent of {Z}, then {(X,Y)} should also be independent of {Z}, and furthermore that

\displaystyle  I(X,Y:Z) \leq I(X:Z) + I(Y:Z)

but these statements are well known to be false (for reasons related to pairwise independence of random variables being strictly weaker than joint independence). For a concrete counterexample, one can take {X, Y \in {\bf F}_2} to be independent, uniformly distributed random elements of the finite field {{\bf F}_2} of two elements, and take {Z := X+Y} to be the sum of these two field elements. One can easily check that each of {X} and {Y} is separately independent of {Z}, but the joint variable {(X,Y)} determines {Z} and thus is not independent of {Z}.

From the inclusion-exclusion identities

\displaystyle  |A \cap C| = |A| + |C| - |A \cup C|

\displaystyle  |B \cap C| = |B| + |C| - |B \cup C|

\displaystyle  |(A \cup B) \cap C| = |A \cup B| + |C| - |A \cup B \cup C|

\displaystyle  |A \cap B \cap C| = |A| + |B| + |C| - |A \cup B| - |B \cup C| - |A \cup C|

\displaystyle + |A \cup B \cup C|

one can check that (1) is equivalent to the trivial lower bound {|A \cap B \cap C| \geq 0}. The basic issue here is that in the dictionary between entropy and combinatorics, there is no satisfactory entropy analogue of the notion of a triple intersection {A \cap B \cap C}. (Even the double intersection {A \cap B} only exists information theoretically in a “virtual” sense; the mutual information {I(X:Y)} allows one to “compute the entropy” of this “intersection”, but does not actually describe this intersection itself as a random variable.)

However, this issue only arises with three or more variables; it is not too difficult to show that the only linear equalities and inequalities that are necessarily obeyed by the information-theoretic quantities {H(X), H(Y), H(X,Y), I(X:Y), H(X|Y), H(Y|X)} associated to just two variables {X,Y} are those that are also necessarily obeyed by their combinatorial analogues {|A|, |B|, |A \cup B|, |A \cap B|, |A \backslash B|, |B \backslash A|}. (See for instance the Venn diagram at the Wikipedia page for mutual information for a pictorial summation of this statement.)

One can work with a larger class of special cases of Shannon entropy by working with random linear functions rather than random boolean functions. Namely, let {S} be some finite-dimensional vector space over a finite field {{\mathbf F}}, and let {f: S \rightarrow {\mathbf F}} be a random linear functional on {S}, selected uniformly among all such functions. Every subspace {U} of {S} then gives rise to a random variable {X = X_U: U \rightarrow {\mathbf F}} formed by restricting {f} to {U}. This random variable is also distributed uniformly amongst all linear functions on {U}, and its entropy can be easily computed to be {\mathrm{dim}(U) \log |\mathbf{F}|}. Given two random variables {X, Y} formed by restricting {f} to {U, V} respectively, the joint random variable {(X,Y)} determines the random linear function {f} on the union {U \cup V} on the two spaces, and thus by linearity on the Minkowski sum {U+V} as well; thus {(X,Y)} is equivalent to the restriction of {f} to {U+V}. In particular, {H(X,Y) = \mathrm{dim}(U+V) \log |\mathbf{F}|}. This implies that {I(X:Y) = \mathrm{dim}(U \cap V) \log |\mathbf{F}|} and also {H(X|Y) = \mathrm{dim}(\pi_V(U)) \log |\mathbf{F}|}, where {\pi_V: S \rightarrow S/V} is the quotient map. After discarding the normalising constant {\log |\mathbf{F}|}, this leads to the following dictionary between information theoretic quantities and linear algebra quantities, analogous to the previous dictionary:

Random variables {X,Y,Z} Subspaces {U,V,W}
Entropy {H(X)} Dimension {\mathrm{dim}(U)}
Joint variable {(X,Y)} Sum {U+V}
Mutual information {I(X:Y)} Dimension of intersection {\mathrm{dim}(U \cap V)}
Conditional entropy {H(X|Y)} Dimension of projection {\mathrm{dim}(\pi_V(U))}
Conditional mutual information {I(X:Y|Z)} {\mathrm{dim}(\pi_W(U) \cap \pi_W(V))}
{X, Y} independent {U, V} transverse ({U \cap V = \{0\}})
{X} determined by {Y} {U} a subspace of {V}
{X,Y} conditionally independent relative to {Z} {\pi_W(U)}, {\pi_W(V)} transverse.

The combinatorial dictionary can be regarded as a specialisation of the linear algebra dictionary, by taking {S} to be the vector space {\mathbf{F}_2^\Omega} over the finite field {\mathbf{F}_2} of two elements, and only considering those subspaces {U} that are coordinate subspaces {U = {\bf F}_2^A} associated to various subsets {A} of {\Omega}.

As before, every linear inequality or equality that is valid for the information-theoretic quantities discussed above, is automatically valid for the linear algebra counterparts for subspaces of a vector space over a finite field by applying the above specialisation (and dividing out by the normalising factor of {\log |\mathbf{F}|}). In fact, the requirement that the field be finite can be removed by applying the compactness theorem from logic (or one of its relatives, such as Los’s theorem on ultraproducts, as done in this previous blog post).

The linear algebra model captures more of the features of Shannon entropy than the combinatorial model. For instance, in contrast to the combinatorial case, it is possible in the linear algebra setting to have subspaces {U,V,W} such that {U} and {V} are separately transverse to {W}, but their sum {U+V} is not; for instance, in a two-dimensional vector space {{\bf F}^2}, one can take {U,V,W} to be the one-dimensional subspaces spanned by {(0,1)}, {(1,0)}, and {(1,1)} respectively. Note that this is essentially the same counterexample from before (which took {{\bf F}} to be the field of two elements). Indeed, one can show that any necessarily true linear inequality or equality involving the dimensions of three subspaces {U,V,W} (as well as the various other quantities on the above table) will also be necessarily true when applied to the entropies of three discrete random variables {X,Y,Z} (as well as the corresponding quantities on the above table).

However, the linear algebra model does not completely capture the subtleties of Shannon entropy once one works with four or more variables (or subspaces). This was first observed by Ingleton, who established the dimensional inequality

\displaystyle  \mathrm{dim}(U \cap V) \leq \mathrm{dim}(\pi_W(U) \cap \pi_W(V)) + \mathrm{dim}(\pi_X(U) \cap \pi_X(V)) + \mathrm{dim}(W \cap X) \ \ \ \ \ (2)

for any subspaces {U,V,W,X}. This is easiest to see when the three terms on the right-hand side vanish; then {\pi_W(U), \pi_W(V)} are transverse, which implies that {U\cap V \subset W}; similarly {U \cap V \subset X}. But {W} and {X} are transverse, and this clearly implies that {U} and {V} are themselves transverse. To prove the general case of Ingleton’s inequality, one can define {Y := U \cap V} and use {\mathrm{dim}(\pi_W(Y)) \leq \mathrm{dim}(\pi_W(U) \cap \pi_W(V))} (and similarly for {X} instead of {W}) to reduce to establishing the inequality

\displaystyle  \mathrm{dim}(Y) \leq \mathrm{dim}(\pi_W(Y)) + \mathrm{dim}(\pi_X(Y)) + \mathrm{dim}(W \cap X) \ \ \ \ \ (3)

which can be rearranged using {\mathrm{dim}(\pi_W(Y)) = \mathrm{dim}(Y) - \mathrm{dim}(W) + \mathrm{dim}(\pi_Y(W))} (and similarly for {X} instead of {W}) and {\mathrm{dim}(W \cap X) = \mathrm{dim}(W) + \mathrm{dim}(X) - \mathrm{dim}(W + X)} as

\displaystyle  \mathrm{dim}(W + X ) \leq \mathrm{dim}(\pi_Y(W)) + \mathrm{dim}(\pi_Y(X)) + \mathrm{dim}(Y)

but this is clear since {\mathrm{dim}(W + X ) \leq \mathrm{dim}(\pi_Y(W) + \pi_Y(X)) + \mathrm{dim}(Y)}.

Returning to the entropy setting, the analogue

\displaystyle  H( V ) \leq H( V | Z ) + H(V | W ) + I(Z:W)

of (3) is true (exercise!), but the analogue

\displaystyle  I(X:Y) \leq I(X:Y|Z) + I(X:Y|W) + I(Z:W) \ \ \ \ \ (4)

of Ingleton’s inequality is false in general. Again, this is easiest to see when all the terms on the right-hand side vanish; then {X,Y} are conditionally independent relative to {Z}, and relative to {W}, and {Z} and {W} are independent, and the claim (4) would then be asserting that {X} and {Y} are independent. While there is no linear counterexample to this statement, there are simple non-linear ones: for instance, one can take {Z,W} to be independent uniform variables from {\mathbf{F}_2}, and take {X} and {Y} to be (say) {ZW} and {(1-Z)(1-W)} respectively (thus {X, Y} are the indicators of the events {Z=W=1} and {Z=W=0} respectively). Once one conditions on either {Z} or {W}, one of {X,Y} has positive conditional entropy and the other has zero entropy, and so {X, Y} are conditionally independent relative to either {Z} or {W}; also, {Z} or {W} are independent of each other. But {X} and {Y} are not independent of each other (they cannot be simultaneously equal to {1}). Somehow, the feature of the linear algebra model that is not present in general is that in the linear algebra setting, every pair of subspaces {U, V} has a well-defined intersection {U \cap V} that is also a subspace, whereas for arbitrary random variables {X, Y}, there does not necessarily exist the analogue of an intersection, namely a “common information” random variable {V} that has the entropy of {I(X:Y)} and is determined either by {X} or by {Y}.

I do not know if there is any simpler model of Shannon entropy that captures all the inequalities available for four variables. One significant complication is that there exist some information inequalities in this setting that are not of Shannon type, such as the Zhang-Yeung inequality

\displaystyle  I(X:Y) \leq 2 I(X:Y|Z) + I(X:Z|Y) + I(Y:Z|X)

\displaystyle + I(X:Y|W) + I(Z:W).

One can however still use these simpler models of Shannon entropy to be able to guess arguments that would work for general random variables. An example of this comes from my paper on the logarithmically averaged Chowla conjecture, in which I showed among other things that

\displaystyle  |\sum_{n \leq x} \frac{\lambda(n) \lambda(n+1)}{n}| \leq \varepsilon x \ \ \ \ \ (5)

whenever {x} was sufficiently large depending on {\varepsilon>0}, where {\lambda} is the Liouville function. The information-theoretic part of the proof was as follows. Given some intermediate scale {H} between {1} and {x}, one can form certain random variables {X_H, Y_H}. The random variable {X_H} is a sign pattern of the form {(\lambda(n+1),\dots,\lambda(n+H))} where {n} is a random number chosen from {1} to {x} (with logarithmic weighting). The random variable {Y_H} was tuple {(n \hbox{ mod } p)_{p \sim \varepsilon^2 H}} of reductions of {n} to primes {p} comparable to {\varepsilon^2 H}. Roughly speaking, what was implicitly shown in the paper (after using the multiplicativity of {\lambda}, the circle method, and the Matomaki-Radziwill theorem on short averages of multiplicative functions) is that if the inequality (5) fails, then there was a lower bound

\displaystyle  I( X_H : Y_H ) \gg \varepsilon^7 \frac{H}{\log H}

on the mutual information between {X_H} and {Y_H}. From translation invariance, this also gives the more general lower bound

\displaystyle  I( X_{H_0,H} : Y_H ) \gg \varepsilon^7 \frac{H}{\log H} \ \ \ \ \ (6)

for any {H_0}, where {X_{H_0,H}} denotes the shifted sign pattern {(\lambda(n+H_0+1),\dots,\lambda(n+H_0+H))}. On the other hand, one had the entropy bounds

\displaystyle  H( X_{H_0,H} ), H(Y_H) \ll H

and from concatenating sign patterns one could see that {X_{H_0,H+H'}} is equivalent to the joint random variable {(X_{H_0,H}, X_{H_0+H,H'})} for any {H_0,H,H'}. Applying these facts and using an “entropy decrement” argument, I was able to obtain a contradiction once {H} was allowed to become sufficiently large compared to {\varepsilon}, but the bound was quite weak (coming ultimately from the unboundedness of {\sum_{\log H_- \leq j \leq \log H_+} \frac{1}{j \log j}} as the interval {[H_-,H_+]} of values of {H} under consideration becomes large), something of the order of {H \sim \exp\exp\exp(\varepsilon^{-7})}; the quantity {H} needs at various junctures to be less than a small power of {\log x}, so the relationship between {x} and {\varepsilon} becomes essentially quadruple exponential in nature, {x \sim \exp\exp\exp\exp(\varepsilon^{-7})}. The basic strategy was to observe that the lower bound (6) causes some slowdown in the growth rate {H(X_{kH})/kH} of the mean entropy, in that this quantity decreased by {\gg \frac{\varepsilon^7}{\log H}} as {k} increased from {1} to {\log H}, basically by dividing {X_{kH}} into {k} components {X_{jH, H}}, {j=0,\dots,k-1} and observing from (6) each of these shares a bit of common information with the same variable {Y_H}. This is relatively clear when one works in a set model, in which {Y_H} is modeled by a set {B_H} of size {O(H)}, and {X_{H_0,H}} is modeled by a set of the form

\displaystyle  X_{H_0,H} = \bigcup_{H_0 < h \leq H_0+H} A_h

for various sets {A_h} of size {O(1)} (also there is some translation symmetry that maps {A_h} to a shift {A_{h+1}} while preserving all of the {B_H}).

However, on considering the set model recently, I realised that one can be a little more efficient by exploiting the fact (basically the Chinese remainder theorem) that the random variables {Y_H} are basically jointly independent as {H} ranges over dyadic values that are much smaller than {\log x}, which in the set model corresponds to the {B_H} all being disjoint. One can then establish a variant

\displaystyle  I( X_{H_0,H} : Y_H | (Y_{H'})_{H' < H}) \gg \varepsilon^7 \frac{H}{\log H} \ \ \ \ \ (7)

of (6), which in the set model roughly speaking asserts that each {B_H} claims a portion of the {\bigcup_{H_0 < h \leq H_0+H} A_h} of cardinality {\gg \varepsilon^7 \frac{H}{\log H}} that is not claimed by previous choices of {B_H}. This leads to a more efficient contradiction (relying on the unboundedness of {\sum_{\log H_- \leq j \leq \log H_+} \frac{1}{j}} rather than {\sum_{\log H_- \leq j \leq \log H_+} \frac{1}{j \log j}}) that looks like it removes one order of exponential growth, thus the relationship between {x} and {\varepsilon} is now {x \sim \exp\exp\exp(\varepsilon^{-7})}. Returning to the entropy model, one can use (7) and Shannon inequalities to establish an inequality of the form

\displaystyle  \frac{1}{2H} H(X_{2H} | (Y_{H'})_{H' \leq 2H}) \leq \frac{1}{H} H(X_{H} | (Y_{H'})_{H' \leq H}) - \frac{c \varepsilon^7}{\log H}

for a small constant {c>0}, which on iterating and using the boundedness of {\frac{1}{H} H(X_{H} | (Y_{H'})_{H' \leq H})} gives the claim. (A modification of this analysis, at least on the level of the back of the envelope calculation, suggests that the Matomaki-Radziwill theorem is needed only for ranges {H} greater than {\exp( (\log\log x)^{\varepsilon^{7}} )} or so, although at this range the theorem is not significantly simpler than the general case).

I’ve just uploaded to the arXiv my paper “Some remarks on the lonely runner conjecture“, submitted to Contributions to discrete mathematics. I had blogged about the lonely runner conjecture in this previous blog post, and I returned to the problem recently to see if I could obtain anything further. The results obtained were more modest than I had hoped, but they did at least seem to indicate a potential strategy to make further progress on the problem, and also highlight some of the difficulties of the problem.

One can rephrase the lonely runner conjecture as the following covering problem. Given any integer “velocity” {v} and radius {0 < \delta < 1/2}, define the Bohr set {B(v,\delta)} to be the subset of the unit circle {{\bf R}/{\bf Z}} given by the formula

\displaystyle B(v,\delta) := \{ t \in {\bf R}/{\bf Z}: \|vt\| \leq \delta \},

where {\|x\|} denotes the distance of {x} to the nearest integer. Thus, for {v} positive, {B(v,\delta)} is simply the union of the {v} intervals {[\frac{a-\delta}{v}, \frac{a+\delta}{v}]} for {a=0,\dots,v-1}, projected onto the unit circle {{\bf R}/{\bf Z}}; in the language of the usual formulation of the lonely runner conjecture, {B(v,\delta)} represents those times in which a runner moving at speed {v} returns to within {\delta} of his or her starting position. For any non-zero integers {v_1,\dots,v_n}, let {\delta(v_1,\dots,v_n)} be the smallest radius {\delta} such that the {n} Bohr sets {B(v_1,\delta),\dots,B(v_n,\delta)} cover the unit circle:

\displaystyle {\bf R}/{\bf Z} = \bigcup_{i=1}^n B(v_i,\delta). \ \ \ \ \ (1)

 

Then define {\delta_n} to be the smallest value of {\delta(v_1,\dots,v_n)}, as {v_1,\dots,v_n} ranges over tuples of distinct non-zero integers. The Dirichlet approximation theorem quickly gives that

\displaystyle \delta(1,\dots,n) = \frac{1}{n+1}

and hence

\displaystyle \delta_n \leq \frac{1}{n+1}

for any {n \geq 1}. The lonely runner conjecture is equivalent to the assertion that this bound is in fact optimal:

Conjecture 1 (Lonely runner conjecture) For any {n \geq 1}, one has {\delta_n = \frac{1}{n+1}}.

This conjecture is currently known for {n \leq 6} (see this paper of Barajas and Serra), but remains open for higher {n}.

It is natural to try to attack the problem by establishing lower bounds on the quantity {\delta_n}. We have the following “trivial” bound, that gets within a factor of two of the conjecture:

Proposition 2 (Trivial bound) For any {n \geq 1}, one has {\delta_n \geq \frac{1}{2n}}.

Proof: It is not difficult to see that for any non-zero velocity {v} and any {0 < \delta < 1/2}, the Bohr set {B(v,\delta)} has Lebesgue measure {m(B(v,\delta)) = 2\delta}. In particular, by the union bound

\displaystyle m(\bigcup_{i=1}^n B(v_i,\delta)) \leq \sum_{i=1}^n m(B(v_i,\delta)) \ \ \ \ \ (2)

 

we see that the covering (1) is only possible if {1 \leq 2 n \delta}, giving the claim. \Box

So, in some sense, all the difficulty is coming from the need to improve upon the trivial union bound (2) by a factor of two.

Despite the crudeness of the union bound (2), it has proven surprisingly hard to make substantial improvements on the trivial bound {\delta_n \geq \frac{1}{2n}}. In 1994, Chen obtained the slight improvement

\displaystyle \delta_n \geq \frac{1}{2n - 1 + \frac{1}{2n-3}}

which was improved a little by Chen and Cusick in 1999 to

\displaystyle \delta_n \geq \frac{1}{2n-3}

when {2n-3} was prime. In a recent paper of Perarnau and Serra, the bound

\displaystyle \delta_n \geq \frac{1}{2n-2+o(1)}

was obtained for arbitrary {n}. These bounds only improve upon the trivial bound by a multiplicative factor of {1+O(1/n)}. Heuristically, one reason for this is as follows. The union bound (2) would of course be sharp if the Bohr sets {B(v_i,\delta)} were all disjoint. Strictly speaking, such disjointness is not possible, because all the Bohr sets {B(v_i,\delta)} have to contain the origin as an interior point. However, it is possible to come up with a large number of Bohr sets {B(v_i,\delta)} which are almost disjoint. For instance, suppose that we had velocities {v_1,\dots,v_s} that were all prime numbers between {n/4} and {n/2}, and that {\delta} was equal to {\delta_n} (and in particular was between {1/2n} and {1/(n+1)}. Then each set {B(v_i,\delta)} can be split into a “kernel” interval {[-\frac{\delta}{v_i}, \frac{\delta}{v_i}]}, together with the “petal” intervals {\bigcup_{a=1}^{v_i-1} [\frac{a-\delta}{v_i}, \frac{a+\delta}{v_i}]}. Roughly speaking, as the prime {v_i} varies, the kernel interval stays more or less fixed, but the petal intervals range over disjoint sets, and from this it is not difficult to show that

\displaystyle m(\bigcup_{i=1}^s B(v_i,\delta)) = (1-O(\frac{1}{n})) \sum_{i=1}^s m(B(v_i,\delta)),

so that the union bound is within a multiplicative factor of {1+O(\frac{1}{n})} of the truth in this case.

This does not imply that {\delta_n} is within a multiplicative factor of {1+O(1/n)} of {\frac{1}{2n}}, though, because there are not enough primes between {n/4} and {n/2} to assign to {n} distinct velocities; indeed, by the prime number theorem, there are only about {\frac{n}{4\log n}} such velocities that could be assigned to a prime. So, while the union bound could be close to tight for up to {\asymp n/\log n} Bohr sets, the above counterexamples don’t exclude improvements to the union bound for larger collections of Bohr sets. Following this train of thought, I was able to obtain a logarithmic improvement to previous lower bounds:

Theorem 3 For sufficiently large {n}, one has {\delta_n \geq \frac{1}{2n} + \frac{c \log n}{n^2 (\log\log n)^2}} for some absolute constant {c>0}.

The factors of {\log\log n} in the denominator are for technical reasons and might perhaps be removable by a more careful argument. However it seems difficult to adapt the methods to improve the {\log n} in the numerator, basically because of the obstruction provided by the near-counterexample discussed above.

Roughly speaking, the idea of the proof of this theorem is as follows. If we have the covering (1) for {\delta} very close to {1/2n}, then the multiplicity function {\sum_{i=1}^n 1_{B(v_i,\delta)}} will then be mostly equal to {1}, but occasionally be larger than {1}. On the other hand, one can compute that the {L^2} norm of this multiplicity function is significantly larger than {1} (in fact it is at least {(3/2-o(1))^{1/2}}). Because of this, the {L^3} norm must be very large, which means that the triple intersections {B(v_i,\delta) \cap B(v_j,\delta) \cap B(v_k,\delta)} must be quite large for many triples {(i,j,k)}. Using some basic Fourier analysis and additive combinatorics, one can deduce from this that the velocities {v_1,\dots,v_n} must have a large structured component, in the sense that there exists an arithmetic progression of length {\asymp n} that contains {\asymp n} of these velocities. For simplicity let us take the arithmetic progression to be {\{1,\dots,n\}}, thus {\asymp n} of the velocities {v_1,\dots,v_n} lie in {\{1,\dots,n\}}. In particular, from the prime number theorem, most of these velocities will not be prime, and will in fact likely have a “medium-sized” prime factor (in the precise form of the argument, “medium-sized” is defined to be “between {\log^{10} n} and {n^{1/10}}“). Using these medium-sized prime factors, one can show that many of the {B(v_i,\delta)} will have quite a large overlap with many of the other {B(v_j,\delta)}, and this can be used after some elementary arguments to obtain a more noticeable improvement on the union bound (2) than was obtained previously.

A modification of the above argument also allows for the improved estimate

\displaystyle \delta(v_1,\dots,v_n) \geq \frac{1+c-o(1)}{2n} \ \ \ \ \ (3)

 

if one knows that all of the velocities {v_1,\dots,v_n} are of size {O(n)}.

In my previous blog post, I showed that in order to prove the lonely runner conjecture, it suffices to do so under the additional assumption that all of the velocities {v_1,\dots,v_n} are of size {O(n^{O(n^2)})}; I reproduce this argument (slightly cleaned up for publication) in the current preprint. There is unfortunately a huge gap between {O(n)} and {O(n^{O(n^2)})}, so the above bound (3) does not immediately give any new bounds for {\delta_n}. However, one could perhaps try to start attacking the lonely runner conjecture by increasing the range {O(n)} for which one has good results, and by decreasing the range {O(n^{O(n^2)})} that one can reduce to. For instance, in the current preprint I give an elementary argument (using a certain amount of case-checking) that shows that the lonely runner bound

\displaystyle \delta(v_1,\dots,v_n) \geq \frac{1}{n+1} \ \ \ \ \ (4)

 

holds if all the velocities {v_1,\dots,v_n} are assumed to lie between {1} and {1.2 n}. This upper threshold of {1.2 n} is only a tiny improvement over the trivial threshold of {n}, but it seems to be an interesting sub-problem of the lonely runner conjecture to increase this threshold further. One key target would be to get up to {2n}, as there are actually a number of {n}-tuples {(v_1,\dots,v_n)} in this range for which (4) holds with equality. The Dirichlet approximation theorem of course gives the tuple {(1,2,\dots,n)}, but there is also the double {(2,4,\dots,2n)} of this tuple, and furthermore there is an additional construction of Goddyn and Wong that gives some further examples such as {(1,2,3,4,5,7,12)}, or more generally one can start with the standard tuple {(1,\dots,n)} and accelerate one of the velocities {v} to {2v}; this turns out to work as long as {v} shares a common factor with every integer between {n-v+1} and {2n-2v+1}. There are a few more examples of this type in the paper of Goddyn and Wong, but all of them can be placed in an arithmetic progression of length {O(n \log n)} at most, so if one were very optimistic, one could perhaps envision a strategy in which the upper bound of {O(n^{O(n^2)})} mentioned earlier was reduced all the way to something like {O( n \log n )}, and then a separate argument deployed to treat this remaining case, perhaps isolating the constructions of Goddyn and Wong (and possible variants thereof) as the only extreme cases.

This is a postscript to the previous blog post which was concerned with obtaining heuristic asymptotic predictions for the correlation

\displaystyle \sum_{n \leq x} \tau(n) \tau(n+h), \ \ \ \ \ (1)

 

for the divisor function {\tau(n) := \sum_{d|n} 1}, in particular recovering the calculation of Ingham that obtained the asymptotic

\displaystyle \sum_{n \leq x} \tau(n) \tau(n+h) \sim \frac{6}{\pi^2} \sigma_{-1}(h) x \log^2 x \ \ \ \ \ (2)

 

when {h} was fixed and non-zero and {x} went to infinity. It is natural to consider the more general correlations

\displaystyle \sum_{n \leq x} \tau_k(n) \tau_l(n+h)

for fixed {k,l \geq 1} and non-zero {h}, where

\displaystyle \tau_k(n) := \sum_{d_1 \dots d_k = n} 1

is the order {k} divisor function. The sum (1) then corresponds to the case {k=l=2}. For {l=1}, {\tau_1(n) = 1}, and a routine application of the Dirichlet hyperbola method (or Perron’s formula) gives the asymptotic

\displaystyle \sum_{n \leq x} \tau_k(n) \sim \frac{\log^{k-1} x}{(k-1)!} x,

or more accurately

\displaystyle \sum_{n \leq x} \tau_k(n) \sim P_k(\log x) x

where {P_k(t)} is a certain explicit polynomial of degree {k-1} with leading coefficient {\frac{1}{(k-1)!}}; see e.g. Exercise 31 of this previous post for a discussion of the {k=3} case (which is already typical). Similarly if {k=1}. For more general {k,l \geq 1}, there is a conjecture of Conrey and Gonek which predicts that

\displaystyle \sum_{n \leq x} \tau_k(n) \tau_l(n+h) \sim P_{k,l,h}(\log x) x

for some polynomial {P_{k,l,h}(t)} of degree {k+l-2} which is explicit but whose form is rather complicated (one has to compute residues of a various complicated products of zeta functions and local factors). This conjecture has been verified when {k \leq 2} or {l \leq 2}, by the work of Linnik, Motohashi, Fouvry-Tenenbaum, and others, but all the remaining cases when {k,l \geq 3} are currently open.

In principle, the calculations of the previous post should recover the predictions of Conrey and Gonek. In this post I would like to record this for the top order term:

Conjecture 1 If {k,l \geq 2} and {h \neq 0} are fixed, then

\displaystyle \sum_{n \leq x} \tau_k(n) \tau_l(n+h) \sim \frac{\log^{k-1} x}{(k-1)!} \frac{\log^{l-1} x}{(l-1)!} x \prod_p {\mathfrak S}_{k,l,p}(h)

as {x \rightarrow \infty}, where the product is over all primes {p}, and the local factors {{\mathfrak S}_{k,l,p}(h)} are given by the formula

\displaystyle {\mathfrak S}_{k,l,p}(h) := (\frac{p-1}{p})^{k+l-2} \sum_{j \geq 0: p^j|h} \frac{1}{p^j} P_{k,l,p}(j) \ \ \ \ \ (3)

 

where {P_{k,l,p}} is the degree {k+l-4} polynomial

\displaystyle P_{k,l,p}(j) := \sum_{k'=2}^k \sum_{l'=2}^l \binom{k-k'+j-1}{k-k'} \binom{l-l'+j-1}{l-l'} \alpha_{k',l',p}

where

\displaystyle \alpha_{k',l',p} := (\frac{p}{p-1})^{k'-1} + (\frac{p}{p-1})^{l'-1} - 1

and one adopts the conventions that {\binom{-1}{0} = 1} and {\binom{m-1}{m} = 0} for {m \geq 1}.

For instance, if {k=l=2} then

\displaystyle P_{2,2,p}(h) = \frac{p}{p-1} + \frac{p}{p-1} - 1 = \frac{p+1}{p-1}

and hence

\displaystyle {\mathfrak S}_{2,2,p}(h) = (1 - \frac{1}{p^2}) \sum_{j \geq 0: p^j|h} \frac{1}{p^j}

and the above conjecture recovers the Ingham formula (2). For {k=2, l=3}, we have

\displaystyle P_{2,3,p}(h) =

\displaystyle (\frac{p}{p-1} + (\frac{p}{p-1})^2 - 1) + (\frac{p}{p-1} + \frac{p}{p-1} - 1) j

\displaystyle = \frac{p^2+p-1}{(p-1)^2} + \frac{p+1}{p-1} j

and so we predict

\displaystyle \sum_{n \leq x} \tau(n) \tau_3(n+h) \sim \frac{x \log^3 x}{2} \prod_p {\mathfrak S}_{2,3,p}(h)

where

\displaystyle {\mathfrak S}_{2,3,p}(h) = \sum_{j \geq 0: p^j|h} \frac{\frac{p^3 - 2p + 1}{p^3} + \frac{(p+1)(p-1)^2}{p^3} j}{p^j}.

Similarly, if {k=l=3} we have

\displaystyle P_{3,3,p}(h) = ((\frac{p}{p-1})^2 + (\frac{p}{p-1})^2 - 1) + 2 (\frac{p}{p-1} + (\frac{p}{p-1})^2 - 1) j

\displaystyle + (\frac{p}{p-1} + \frac{p}{p-1} - 1) j^2

\displaystyle = \frac{p^2+2p-1}{(p-1)^2} + 2 \frac{p^2+p-1}{(p-1)^2} j + \frac{p+1}{p-1} j^2

and so we predict

\displaystyle \sum_{n \leq x} \tau_3(n) \tau_3(n+h) \sim \frac{x \log^4 x}{4} \prod_p {\mathfrak S}_{3,3,p}(h)

where

\displaystyle {\mathfrak S}_{3,3,p}(h) = \sum_{j \geq 0: p^j|h} \frac{\frac{p^4 - 4p^2 + 4p - 1}{p^4} + 2 \frac{(p^2+p-1)(p-1)^2}{p^4} j + \frac{(p+1)(p-1)^3}{p^4} j^2}{p^j}.

and so forth.

As in the previous blog, the idea is to factorise

\displaystyle \tau_k(n) = \prod_p \tau_{k,p}(n)

where the local factors {\tau_{k,p}(n)} are given by

\displaystyle \tau_{k,p}(n) := \sum_{j_1,\dots,j_k \geq 0: p^{j_1+\dots+j_k} || n} 1

(where {p^j || n} means that {p} divides {n} precisely {j} times), or in terms of the valuation {v_p(n)} of {n} at {p},

\displaystyle \tau_{k,p}(n) = \binom{k-1+v_p(n)}{k-1}. \ \ \ \ \ (4)

 

We then have the following exact local asymptotics:

Proposition 2 (Local correlations) Let {{\bf n}} be a profinite integer chosen uniformly at random, let {h} be a profinite integer, and let {k,l \geq 2}. Then

\displaystyle {\bf E} \tau_{k,p}({\bf n}) = (\frac{p}{p-1})^{k-1} \ \ \ \ \ (5)

 

and

\displaystyle {\bf E} \tau_{k,p}({\bf n}) \tau_{l,p}({\bf n}+h) = (\frac{p}{p-1})^{k+l-2} {\mathfrak S}_{k,l,p}(h). \ \ \ \ \ (6)

 

(For profinite integers it is possible that {v_p({\bf n})} and hence {\tau_{k,p}({\bf n})} are infinite, but this is a probability zero event and so can be ignored.)

Conjecture 1 can then be heuristically justified from the local calculations (2) by various pseudorandomness heuristics, as discussed in the previous post.

I’ll give a short proof of the above proposition below, basically using the recursive methods of the previous post. This short proof actually took be quite a while to find; I spent several hours and a fair bit of scratch paper working out the cases {k,l = 2,3} laboriously by hand (with some assistance and cross-checking from Maple). Here is an unorganised sample of some of this scratch, just to show how the sausage is actually made:

WP_20160831_12_53_59_Pro

It was only after expending all this effort that I realised that it would be much more efficient to compute the correlations for all values of {k,l} simultaneously by using generating functions. After performing this computation, it then became apparent that there would be a direct combinatorial proof of (6) that was shorter than even the generating function proof. (I will not supply the full generating function calculations here, but will at least show them for the easier correlation (5).)

I am confident that Conjecture 1 is consistent with the explicit asymptotic in the Conrey-Gonek conjecture, but have not yet rigorously established that the leading order term in the latter is indeed identical to the expression provided above.

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Let {\tau(n) := \sum_{d|n} 1} be the divisor function. A classical application of the Dirichlet hyperbola method gives the asymptotic

\displaystyle \sum_{n \leq x} \tau(n) \sim x \log x

where {X \sim Y} denotes the estimate {X = (1+o(1))Y} as {x \rightarrow \infty}. Much better error estimates are possible here, but we will not focus on the lower order terms in this discussion. For somewhat idiosyncratic reasons I will interpret this estimate (and the other analytic number theory estimates discussed here) through the probabilistic lens. Namely, if {{\bf n} = {\bf n}_x} is a random number selected uniformly between {1} and {x}, then the above estimate can be written as

\displaystyle {\bf E} \tau( {\bf n} ) \sim \log x, \ \ \ \ \ (1)

 

that is to say the random variable {\tau({\bf n})} has mean approximately {\log x}. (But, somewhat paradoxically, this is not the median or mode behaviour of this random variable, which instead concentrates near {\log^{\log 2} x}, basically thanks to the Hardy-Ramanujan theorem.)

Now we turn to the pair correlations {\sum_{n \leq x} \tau(n) \tau(n+h)} for a fixed positive integer {h}. There is a classical computation of Ingham that shows that

\displaystyle \sum_{n \leq x} \tau(n) \tau(n+h) \sim \frac{6}{\pi^2} \sigma_{-1}(h) x \log^2 x, \ \ \ \ \ (2)

 

where

\displaystyle \sigma_{-1}(h) := \sum_{d|h} \frac{1}{d}.

The error term in (2) has been refined by many subsequent authors, as has the uniformity of the estimates in the {h} aspect, as these topics are related to other questions in analytic number theory, such as fourth moment estimates for the Riemann zeta function; but we will not consider these more subtle features of the estimate here. However, we will look at the next term in the asymptotic expansion for (2) below the fold.

Using our probabilistic lens, the estimate (2) can be written as

\displaystyle {\bf E} \tau( {\bf n} ) \tau( {\bf n} + h ) \sim \frac{6}{\pi^2} \sigma_{-1}(h) \log^2 x. \ \ \ \ \ (3)

 

From (1) (and the asymptotic negligibility of the shift by {h}) we see that the random variables {\tau({\bf n})} and {\tau({\bf n}+h)} both have a mean of {\sim \log x}, so the additional factor of {\frac{6}{\pi^2} \sigma_{-1}(h)} represents some arithmetic coupling between the two random variables.

Ingham’s formula can be established in a number of ways. Firstly, one can expand out {\tau(n) = \sum_{d|n} 1} and use the hyperbola method (splitting into the cases {d \leq \sqrt{x}} and {n/d \leq \sqrt{x}} and removing the overlap). If one does so, one soon arrives at the task of having to estimate sums of the form

\displaystyle \sum_{n \leq x: d|n} \tau(n+h)

for various {d \leq \sqrt{x}}. For {d} much less than {\sqrt{x}} this can be achieved using a further application of the hyperbola method, but for {d} comparable to {\sqrt{x}} things get a bit more complicated, necessitating the use of non-trivial estimates on Kloosterman sums in order to obtain satisfactory control on error terms. A more modern approach proceeds using automorphic form methods, as discussed in this previous post. A third approach, which unfortunately is only heuristic at the current level of technology, is to apply the Hardy-Littlewood circle method (discussed in this previous post) to express (2) in terms of exponential sums {\sum_{n \leq x} \tau(n) e(\alpha n)} for various frequencies {\alpha}. The contribution of “major arc” {\alpha} can be computed after a moderately lengthy calculation which yields the right-hand side of (2) (as well as the correct lower order terms that are currently being suppressed), but there does not appear to be an easy way to show directly that the “minor arc” contributions are of lower order, although the methods discussed previously do indirectly show that this is ultimately the case.

Each of the methods outlined above requires a fair amount of calculation, and it is not obvious while performing them that the factor {\frac{6}{\pi^2} \sigma_{-1}(h)} will emerge at the end. One can at least explain the {\frac{6}{\pi^2}} as a normalisation constant needed to balance the {\sigma_{-1}(h)} factor (at a heuristic level, at least). To see this through our probabilistic lens, introduce an independent copy {{\bf n}'} of {{\bf n}}, then

\displaystyle {\bf E} \tau( {\bf n} ) \tau( {\bf n}' ) = ({\bf E} \tau ({\bf n}))^2 \sim \log^2 x; \ \ \ \ \ (4)

 

using symmetry to order {{\bf n}' > {\bf n}} (discarding the diagonal case {{\bf n} = {\bf n}'}) and making the change of variables {{\bf n}' = {\bf n}+h}, we see that (4) is heuristically consistent with (3) as long as the asymptotic mean of {\frac{6}{\pi^2} \sigma_{-1}(h)} in {h} is equal to {1}. (This argument is not rigorous because there was an implicit interchange of limits present, but still gives a good heuristic “sanity check” of Ingham’s formula.) Indeed, if {{\bf E}_h} denotes the asymptotic mean in {h}, then we have (heuristically at least)

\displaystyle {\bf E}_h \sigma_{-1}(h) = \sum_d {\bf E}_h \frac{1}{d} 1_{d|h}

\displaystyle = \sum_d \frac{1}{d^2}

\displaystyle = \frac{\pi^2}{6}

and we obtain the desired consistency after multiplying by {\frac{6}{\pi^2}}.

This still however does not explain the presence of the {\sigma_{-1}(h)} factor. Intuitively it is reasonable that if {h} has many prime factors, and {{\bf n}} has a lot of factors, then {{\bf n}+h} will have slightly more factors than average, because any common factor to {h} and {{\bf n}} will automatically be acquired by {{\bf n}+h}. But how to quantify this effect?

One heuristic way to proceed is through analysis of local factors. Observe from the fundamental theorem of arithmetic that we can factor

\displaystyle \tau(n) = \prod_p \tau_p(n)

where the product is over all primes {p}, and {\tau_p(n) := \sum_{p^j|n} 1} is the local version of {\tau(n)} at {p} (which in this case, is just one plus the {p}valuation {v_p(n)} of {n}: {\tau_p = 1 + v_p}). Note that all but finitely many of the terms in this product will equal {1}, so the infinite product is well-defined. In a similar fashion, we can factor

\displaystyle \sigma_{-1}(h) = \prod_p \sigma_{-1,p}(h)

where

\displaystyle \sigma_{-1,p}(h) := \sum_{p^j|h} \frac{1}{p^j}

(or in terms of valuations, {\sigma_{-1,p}(h) = (1 - p^{-v_p(h)-1})/(1-p^{-1})}). Heuristically, the Chinese remainder theorem suggests that the various factors {\tau_p({\bf n})} behave like independent random variables, and so the correlation between {\tau({\bf n})} and {\tau({\bf n}+h)} should approximately decouple into the product of correlations between the local factors {\tau_p({\bf n})} and {\tau_p({\bf n}+h)}. And indeed we do have the following local version of Ingham’s asymptotics:

Proposition 1 (Local Ingham asymptotics) For fixed {p} and integer {h}, we have

\displaystyle {\bf E} \tau_p({\bf n}) \sim \frac{p}{p-1}

and

\displaystyle {\bf E} \tau_p({\bf n}) \tau_p({\bf n}+h) \sim (1-\frac{1}{p^2}) \sigma_{-1,p}(h) (\frac{p}{p-1})^2

\displaystyle = \frac{p+1}{p-1} \sigma_{-1,p}(h)

From the Euler formula

\displaystyle \prod_p (1-\frac{1}{p^2}) = \frac{1}{\zeta(2)} = \frac{6}{\pi^2}

we see that

\displaystyle \frac{6}{\pi^2} \sigma_{-1}(h) = \prod_p (1-\frac{1}{p^2}) \sigma_{-1,p}(h)

and so one can “explain” the arithmetic factor {\frac{6}{\pi^2} \sigma_{-1}(h)} in Ingham’s asymptotic as the product of the arithmetic factors {(1-\frac{1}{p^2}) \sigma_{-1,p}(h)} in the (much easier) local Ingham asymptotics. Unfortunately we have the usual “local-global” problem in that we do not know how to rigorously derive the global asymptotic from the local ones; this problem is essentially the same issue as the problem of controlling the minor arc contributions in the circle method, but phrased in “physical space” language rather than “frequency space”.

Remark 2 The relation between the local means {\sim \frac{p}{p-1}} and the global mean {\sim \log^2 x} can also be seen heuristically through the application

\displaystyle \prod_{p \leq x^{1/e^\gamma}} \frac{p}{p-1} \sim \log x

of Mertens’ theorem, where {1/e^\gamma} is Pólya’s magic exponent, which serves as a useful heuristic limiting threshold in situations where the product of local factors is divergent.

Let us now prove this proposition. One could brute-force the computations by observing that for any fixed {j}, the valuation {v_p({\bf n})} is equal to {j} with probability {\sim \frac{p-1}{p} \frac{1}{p^j}}, and with a little more effort one can also compute the joint distribution of {v_p({\bf n})} and {v_p({\bf n}+h)}, at which point the proposition reduces to the calculation of various variants of the geometric series. I however find it cleaner to proceed in a more recursive fashion (similar to how one can prove the geometric series formula by induction); this will also make visible the vague intuition mentioned previously about how common factors of {{\bf n}} and {h} force {{\bf n}+h} to have a factor also.

It is first convenient to get rid of error terms by observing that in the limit {x \rightarrow \infty}, the random variable {{\bf n} = {\bf n}_x} converges vaguely to a uniform random variable {{\bf n}_\infty} on the profinite integers {\hat {\bf Z}}, or more precisely that the pair {(v_p({\bf n}_x), v_p({\bf n}_x+h))} converges vaguely to {(v_p({\bf n}_\infty), v_p({\bf n}_\infty+h))}. Because of this (and because of the easily verified uniform integrability properties of {\tau_p({\bf n})} and their powers), it suffices to establish the exact formulae

\displaystyle {\bf E} \tau_p({\bf n}_\infty) = \frac{p}{p-1} \ \ \ \ \ (5)

 

and

\displaystyle {\bf E} \tau_p({\bf n}_\infty) \tau_p({\bf n}_\infty+h) = (1-\frac{1}{p^2}) \sigma_{-1,p}(h) (\frac{p}{p-1})^2 = \frac{p+1}{p-1} \sigma_{-1,p}(h) \ \ \ \ \ (6)

 

in the profinite setting (this setting will make it easier to set up the recursion).

We begin with (5). Observe that {{\bf n}_\infty} is coprime to {p} with probability {\frac{p-1}{p}}, in which case {\tau_p({\bf n}_\infty)} is equal to {1}. Conditioning to the complementary probability {\frac{1}{p}} event that {{\bf n}_\infty} is divisible by {p}, we can factor {{\bf n}_\infty = p {\bf n}'_\infty} where {{\bf n}'_\infty} is also uniformly distributed over the profinite integers, in which event we have {\tau_p( {\bf n}_\infty ) = 1 + \tau_p( {\bf n}'_\infty )}. We arrive at the identity

\displaystyle {\bf E} \tau_p({\bf n}_\infty) = \frac{p-1}{p} + \frac{1}{p} ( 1 + {\bf E} \tau_p( {\bf n}'_\infty ) ).

As {{\bf n}_\infty} and {{\bf n}'_\infty} have the same distribution, the quantities {{\bf E} \tau_p({\bf n}_\infty)} and {{\bf E} \tau_p({\bf n}'_\infty)} are equal, and (5) follows by a brief amount of high-school algebra.

We use a similar method to treat (6). First treat the case when {h} is coprime to {p}. Then we see that with probability {\frac{p-2}{p}}, {{\bf n}_\infty} and {{\bf n}_\infty+h} are simultaneously coprime to {p}, in which case {\tau_p({\bf n}_\infty) = \tau_p({\bf n}_\infty+h) = 1}. Furthermore, with probability {\frac{1}{p}}, {{\bf n}_\infty} is divisible by {p} and {{\bf n}_\infty+h} is not; in which case we can write {{\bf n} = p {\bf n}'} as before, with {\tau_p({\bf n}_\infty) = 1 + \tau_p({\bf n}'_\infty)} and {\tau_p({\bf n}_\infty+h)=1}. Finally, in the remaining event with probability {\frac{1}{p}}, {{\bf n}+h} is divisible by {p} and {{\bf n}} is not; we can then write {{\bf n}_\infty+h = p {\bf n}'_\infty}, so that {\tau_p({\bf n}_\infty+h) = 1 + \tau_p({\bf n}'_\infty)} and {\tau_p({\bf n}_\infty) = 1}. Putting all this together, we obtain

\displaystyle {\bf E} \tau_p({\bf n}_\infty) \tau_p({\bf n}_\infty+h) = \frac{p-2}{p} + 2 \frac{1}{p} (1 + {\bf E} \tau_p({\bf n}'_\infty))

and the claim (6) in this case follows from (5) and a brief computation (noting that {\sigma_{-1,p}(h)=1} in this case).

Now suppose that {h} is divisible by {p}, thus {h=ph'} for some integer {h'}. Then with probability {\frac{p-1}{p}}, {{\bf n}_\infty} and {{\bf n}_\infty+h} are simultaneously coprime to {p}, in which case {\tau_p({\bf n}_\infty) = \tau_p({\bf n}_\infty+h) = 1}. In the remaining {\frac{1}{p}} event, we can write {{\bf n}_\infty = p {\bf n}'_\infty}, and then {\tau_p({\bf n}_\infty) = 1 + \tau_p({\bf n}'_\infty)} and {\tau_p({\bf n}_\infty+h) = 1 + \tau_p({\bf n}'_\infty+h')}. Putting all this together we have

\displaystyle {\bf E} \tau_p({\bf n}_\infty) \tau_p({\bf n}_\infty+h) = \frac{p-1}{p} + \frac{1}{p} {\bf E} (1+\tau_p({\bf n}'_\infty)(1+\tau_p({\bf n}'_\infty+h)

which by (5) (and replacing {{\bf n}'_\infty} by {{\bf n}_\infty}) leads to the recursive relation

\displaystyle {\bf E} \tau_p({\bf n}_\infty) \tau_p({\bf n}_\infty+h) = \frac{p+1}{p-1} + \frac{1}{p} {\bf E} \tau_p({\bf n}_\infty) \tau_p({\bf n}_\infty+h)

and (6) then follows by induction on the number of powers of {p}.

The estimate (2) of Ingham was refined by Estermann, who obtained the more accurate expansion

\displaystyle \sum_{n \leq x} \tau(n) \tau(n+h) = \frac{6}{\pi^2} \sigma_{-1}(h) x \log^2 x + a_1(h) x \log x + a_2(h) x \ \ \ \ \ (7)

 

\displaystyle + O( x^{11/12+o(1)} )

for certain complicated but explicit coefficients {a_1(h), a_2(h)}. For instance, {a_1(h)} is given by the formula

\displaystyle a_1(h) = (\frac{12}{\pi^2} (2\gamma-1) + 4 a') \sigma_{-1}(h) - \frac{24}{\pi^2} \sigma'_{-1}(h)

where {\gamma} is the Euler-Mascheroni constant,

\displaystyle a' := - \sum_{r=1}^\infty \frac{\mu(r)}{r^2} \log r, \ \ \ \ \ (8)

 

and

\displaystyle \sigma'_{-1}(h) := \sum_{d|h} \frac{\log d}{d}.

The formula for {a_2(h)} is similar but even more complicated. The error term {O( x^{11/12+o(1)})} was improved by Heath-Brown to {O( x^{5/6+o(1)})}; it is conjectured (for instance by Conrey and Gonek) that one in fact has square root cancellation {O( x^{1/2+o(1)})} here, but this is well out of reach of current methods.

These lower order terms are traditionally computed either from a Dirichlet series approach (using Perron’s formula) or a circle method approach. It turns out that a refinement of the above heuristics can also predict these lower order terms, thus keeping the calculation purely in physical space as opposed to the “multiplicative frequency space” of the Dirichlet series approach, or the “additive frequency space” of the circle method, although the computations are arguably as messy as the latter computations for the purposes of working out the lower order terms. We illustrate this just for the {a_1(h) x \log x} term below the fold.

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The twin prime conjecture, still unsolved, asserts that there are infinitely many primes {p} such that {p+2} is also prime. A more precise form of this conjecture is (a special case) of the Hardy-Littlewood prime tuples conjecture, which asserts that

\displaystyle \sum_{n \leq x} \Lambda(n) \Lambda(n+2) = (2\Pi_2+o(1)) x \ \ \ \ \ (1)

 

as {x \rightarrow \infty}, where {\Lambda} is the von Mangoldt function and {\Pi_2 = 0.6606\dots} is the twin prime constant

\displaystyle \prod_{p>2} (1 - \frac{1}{(p-1)^2}).

Because {\Lambda} is almost entirely supported on the primes, it is not difficult to see that (1) implies the twin prime conjecture.

One can give a heuristic justification of the asymptotic (1) (and hence the twin prime conjecture) via sieve theoretic methods. Recall that the von Mangoldt function can be decomposed as a Dirichlet convolution

\displaystyle \Lambda(n) = \sum_{d|n} \mu(d) \log \frac{n}{d}

where {\mu} is the Möbius function. Because of this, we can rewrite the left-hand side of (1) as

\displaystyle \sum_{d \leq x} \mu(d) \sum_{n \leq x: d|n} \log\frac{n}{d} \Lambda(n+2). \ \ \ \ \ (2)

 

To compute this double sum, it is thus natural to consider sums such as

\displaystyle \sum_{n \leq x: d|n} \log \frac{n}{d} \Lambda(n+2)

or (to simplify things by removing the logarithm)

\displaystyle \sum_{n \leq x: d|n} \Lambda(n+2).

The prime number theorem in arithmetic progressions suggests that one has an asymptotic of the form

\displaystyle \sum_{n \leq x: d|n} \Lambda(n+2) \approx \frac{g(d)}{d} x \ \ \ \ \ (3)

 

where {g} is the multiplicative function with {g(d)=0} for {d} even and

\displaystyle g(d) := \frac{d}{\phi(d)} = \prod_{p|d} (1-\frac{1}{p})^{-1}

for {d} odd. Summing by parts, one then expects

\displaystyle \sum_{n \leq x: d|n} \Lambda(n+2)\log \frac{n}{d}  \approx \frac{g(d)}{d} x \log \frac{x}{d}

and so we heuristically have

\displaystyle \sum_{n \leq x} \Lambda(n) \Lambda(n+2) \approx x \sum_{d \leq x} \frac{\mu(d) g(d)}{d} \log \frac{x}{d}.

The Dirichlet series

\displaystyle \sum_n \frac{\mu(n) g(n)}{n^s}

has an Euler product factorisation

\displaystyle \sum_n \frac{\mu(n) g(n)}{n^s} = \prod_p (1 - \frac{g(p)}{p^s})

for {\hbox{Re} s > 1}; comparing this with the Euler product factorisation

\displaystyle \zeta(s) = \prod_p (1 - \frac{1}{p^s})^{-1}

for the Riemann zeta function, and recalling that {\zeta} has a simple pole of residue {1} at {s=1}, we see that

\displaystyle \sum_n \frac{\mu(n) g(n)}{n^s} = \frac{1}{\zeta(s)} \prod_p \frac{1-g(p)/p^s}{1-p^s}

has a simple zero at {s=1} with first derivative

\displaystyle \prod_p \frac{1 - g(p)/p}{1-1/p} = 2 \Pi_2.

From this and standard multiplicative number theory manipulations, one can calculate the asymptotic

\displaystyle \sum_{d \leq x} \frac{\mu(d) g(d)}{d} \log \frac{x}{d} = 2 \Pi_2 + o(1)

which concludes the heuristic justification of (1).

What prevents us from making the above heuristic argument rigorous, and thus proving (1) and the twin prime conjecture? Note that the variable {d} in (2) ranges to be as large as {x}. On the other hand, the prime number theorem in arithmetic progressions (3) is not expected to hold for {d} anywhere that large (for instance, the left-hand side of (3) vanishes as soon as {d} exceeds {x}). The best unconditional result known of the type (3) is the Siegel-Walfisz theorem, which allows {d} to be as large as {\log^{O(1)} x}. Even the powerful generalised Riemann hypothesis (GRH) only lets one prove an estimate of the form (3) for {d} up to about {x^{1/2-o(1)}}.

However, because of the averaging effect of the summation in {d} in (2), we don’t need the asymptotic (3) to be true for all {d} in a particular range; having it true for almost all {d} in that range would suffice. Here the situation is much better; the celebrated Bombieri-Vinogradov theorem (sometimes known as “GRH on the average”) implies, roughly speaking, that the approximation (3) is valid for almost all {d \leq x^{1/2-\varepsilon}} for any fixed {\varepsilon>0}. While this is not enough to control (2) or (1), the Bombieri-Vinogradov theorem can at least be used to control variants of (1) such as

\displaystyle \sum_{n \leq x} (\sum_{d|n} \lambda_d) \Lambda(n+2)

for various sieve weights {\lambda_d} whose associated divisor function {\sum_{d|n} \lambda_d} is supposed to approximate the von Mangoldt function {\Lambda}, although that theorem only lets one do this when the weights {\lambda_d} are supported on the range {d \leq x^{1/2-\varepsilon}}. This is still enough to obtain some partial results towards (1); for instance, by selecting weights according to the Selberg sieve, one can use the Bombieri-Vinogradov theorem to establish the upper bound

\displaystyle \sum_{n \leq x} \Lambda(n) \Lambda(n+2) \leq (4+o(1)) 2 \Pi_2 x, \ \ \ \ \ (4)

 

which is off from (1) by a factor of about {4}. See for instance this blog post for details.

It has been difficult to improve upon the Bombieri-Vinogradov theorem in its full generality, although there are various improvements to certain restricted versions of the Bombieri-Vinogradov theorem, for instance in the famous work of Zhang on bounded gaps between primes. Nevertheless, it is believed that the Elliott-Halberstam conjecture (EH) holds, which roughly speaking would mean that (3) now holds for almost all {d \leq x^{1-\varepsilon}} for any fixed {\varepsilon>0}. (Unfortunately, the {\varepsilon} factor cannot be removed, as investigated in a series of papers by Friedlander, Granville, and also Hildebrand and Maier.) This comes tantalisingly close to having enough distribution to control all of (1). Unfortunately, it still falls short. Using this conjecture in place of the Bombieri-Vinogradov theorem leads to various improvements to sieve theoretic bounds; for instance, the factor of {4+o(1)} in (4) can now be improved to {2+o(1)}.

In two papers from the 1970s (which can be found online here and here respectively, the latter starting on page 255 of the pdf), Bombieri developed what is now known as the Bombieri asymptotic sieve to clarify the situation more precisely. First, he showed that on the Elliott-Halberstam conjecture, while one still could not establish the asymptotic (1), one could prove the generalised asymptotic

\displaystyle \sum_{n \leq x} \Lambda_k(n) \Lambda(n+2) = (2\Pi_2+o(1)) k x \log^{k-1} x \ \ \ \ \ (5)

 

for all natural numbers {k \geq 2}, where the generalised von Mangoldt functions {\Lambda_k} are defined by the formula

\displaystyle \Lambda_k(n) := \sum_{d|n} \mu(d) \log^k \frac{n}{d}.

These functions behave like the von Mangoldt function, but are concentrated on {k}-almost primes (numbers with at most {k} prime factors) rather than primes. The right-hand side of (5) corresponds to what one would expect if one ran the same heuristics used to justify (1). Sadly, the {k=1} case of (5), which is just (1), is just barely excluded from Bombieri’s analysis.

More generally, on the assumption of EH, the Bombieri asymptotic sieve provides the asymptotic

\displaystyle \sum_{n \leq x} \Lambda_{(k_1,\dots,k_r)}(n) \Lambda(n+2) \ \ \ \ \ (6)

 

\displaystyle = (2\Pi_2+o(1)) \frac{\prod_{i=1}^r k_i!}{(k_1+\dots+k_r-1)!} x \log^{k_1+\dots+k_r-1} x

for any fixed {r \geq 1} and any tuple {(k_1,\dots,k_r)} of natural numbers other than {(1,\dots,1)}, where

\displaystyle \Lambda_{(k_1,\dots,k_r)} := \Lambda_{k_1} * \dots * \Lambda_{k_r}

is a further generalisation of the von Mangoldt function (now concentrated on {k_1+\dots+k_r}-almost primes). By combining these asymptotics with some elementary identities involving the {\Lambda_{(k_1,\dots,k_r)}}, together with the Weierstrass approximation theorem, Bombieri was able to control a wide family of sums including (1), except for one undetermined scalar {\delta_x \in [0,2]}. Namely, he was able to show (again on EH) that for any fixed {r \geq 1} and any continuous function {g_r} on the simplex {\Delta_r := \{ (t_1,\dots,t_r) \in {\bf R}^r: t_1+\dots+t_r = 1; 0 \leq t_1 \leq \dots \leq t_r\}} that had suitable vanishing at the boundary, the sum

\displaystyle \sum_{n \leq x: n=p_1 \dots p_r} g_r( \frac{\log p_1}{\log n}, \dots, \frac{\log p_r}{\log n} ) \Lambda(n+2)

was equal to

\displaystyle (\delta_x+o(1)) \int_{\Delta_r} g_r \frac{x}{\log x} \ \ \ \ \ (7)

 

when {r} was odd and

\displaystyle (2-\delta_x+o(1)) \int_{\Delta_r} g_r \frac{x}{\log x} \ \ \ \ \ (8)

 

when {r} was even, where the integral on {\Delta_r} is with respect to the measure {\frac{dt_1 \dots dt_{r-1}}{t_1 \dots t_r}} (this is Dirac measure in the case {r=1}). In particular, we have

\displaystyle \sum_{n \leq x} \Lambda(n) \Lambda(n+2) = (\delta_x + o(1)) 2 \Pi_2 x

and the twin prime conjecture would be proved if one could show that {\delta_x} is bounded away from zero, while (1) is equivalent to the assertion that {\delta_x} is equal to {1+o(1)}. Unfortunately, no additional bound beyond the inequalities {0 \leq \delta_x \leq 2} provided by the Bombieri asymptotic sieve is known, even if one assumes all other major conjectures in number theory than the prime tuples conjecture and its variants (e.g. GRH, GEH, GUE, abc, Chowla, …).

To put it another way, the Bombieri asymptotic sieve is able (on EH) to compute asymptotics for sums

\displaystyle \sum_{n \leq x} f(n) \Lambda(n+2) \ \ \ \ \ (9)

 

without needing to know the unknown scalar {\delta_x}, when {f} is a function supported on almost primes of the form

\displaystyle f(p_1 \dots p_r) = g_r( \frac{\log p_1}{\log n}, \dots, \frac{\log p_r}{\log n} )

for {1 \leq r \leq r_*} and some fixed {r_*}, with {f} vanishing elsewhere and for some continuous (symmetric) functions {g_r: \Delta_r \rightarrow {\bf C}} obeying some vanishing at the boundary, so long as the parity condition

\displaystyle \sum_{r \hbox{ odd}} \int_{\Delta_r} g_r = \sum_{r \hbox{ even}} \int_{\Delta_r} g_r

is obeyed (informally: {f} gives the same weight to products of an odd number of primes as to products of an even number of primes, or to put it another way, {f} is asymptotically orthogonal to the Möbius function {\mu}). But when {f} violates the parity condition, the asymptotic involves the unknown {\delta_x}. This scalar {\delta_x} thus embodies the “parity problem” for the twin prime conjecture (discussed in these previous blog posts).

Because the obstruction to the parity problem is only one-dimensional (on EH), one can replace any parity-violating weight (such as {\Lambda}) with any other parity-violating weight and obtain a logically equivalent estimate. For instance, to prove the twin prime conjecture on EH, it would suffice to show that

\displaystyle \sum_{p_1 p_2 p_3 \leq x: p_1,p_2,p_3 \geq x^\alpha} \Lambda(p_1 p_2 p_3 + 2) \gg \frac{x}{\log x}

for some fixed {\alpha>0}, or equivalently that there are {\gg \frac{x}{\log^2 x}} solutions to the equation {p - p_1 p_2 p_3 = 2} in primes with {p \leq x} and {p_1,p_2,p_3 \geq x^\alpha}. (In some cases, this sort of reduction can also be made using other sieves than the Bombieri asymptotic sieve, as was observed by Ng.) As another example, the Bombieri asymptotic sieve can be used to show that the asymptotic (1) is equivalent to the asymptotic

\displaystyle \sum_{n \leq x} \mu(n) 1_R(n) \Lambda(n+2) = o( \frac{x}{\log x})

where {R} is the set of numbers that are rough in the sense that they have no prime factors less than {x^\alpha} for some fixed {\alpha>0} (the function {\mu 1_R} clearly correlates with {\mu} and so must violate the parity condition). One can replace {1_R} with similar sieve weights (e.g. a Selberg sieve) that concentrate on almost primes if desired.

As it turns out, if one is willing to strengthen the assumption of the Elliott-Halberstam (EH) conjecture to the assumption of the generalised Elliott-Halberstam (GEH) conjecture (as formulated for instance in Claim 2.6 of the Polymath8b paper), one can also swap the {\Lambda(n+2)} factor in the above asymptotics with other parity-violating weights and obtain a logically equivalent estimate, as the Bombieri asymptotic sieve also applies to weights such as {\mu 1_R} under the assumption of GEH. For instance, on GEH one can use two such applications of the Bombieri asymptotic sieve to show that the twin prime conjecture would follow if one could show that there are {\gg \frac{x}{\log^2 x}} solutions to the equation

\displaystyle p_1 p_2 - p_3 p_4 = 2

in primes with {p_1,p_2,p_3,p_4 \geq x^\alpha} and {p_1 p_2 \leq x}, for some {\alpha > 0}. Similarly, on GEH the asymptotic (1) is equivalent to the asymptotic

\displaystyle \sum_{n \leq x} \mu(n) 1_R(n) \mu(n+2) 1_R(n+2) = o( \frac{x}{\log^2 x})

for some fixed {\alpha>0}, and similarly with {1_R} replaced by other sieves. This form of the quantitative twin primes conjecture is appealingly similar to the (special case)

\displaystyle \sum_{n \leq x} \mu(n) \mu(n+2) = o(x)

of the Chowla conjecture, for which there has been some recent progress (discussed for instance in these recent posts). Informally, the Bombieri asymptotic sieve lets us (on GEH) view the twin prime conjecture as a sort of Chowla conjecture restricted to almost primes. Unfortunately, the recent progress on the Chowla conjecture relies heavily on the multiplicativity of {\mu} at small primes, which is completely destroyed by inserting a weight such as {1_R}, so this does not yet yield a viable path towards the twin prime conjecture even assuming GEH. Still, the similarity is striking, and one can hope that further ways to attack the Chowla conjecture may emerge that could impact the twin prime conjecture. (Alternatively, if one assumes a sufficiently optimistic version of the GEH, one could perhaps relax the notion of “almost prime” to the extent that one could start usefully using multiplicativity at smallish primes, though this seems rather wishful at present, particularly since the most optimistic versions of GEH are known to be false.)

The Bombieri asymptotic sieve is already well explained in the original two papers of Bombieri; there is also a slightly different treatment of the sieve by Friedlander and Iwaniec, as well as a simplified version in the book of Friedlander and Iwaniec (in which the distribution hypothesis is strengthened in order to shorten the arguments. I’ve decided though to write up my own notes on the sieve below the fold; this is primarily for my own benefit, but may be useful to some readers also. I largely follow the treatment of Bombieri, with the one idiosyncratic twist of replacing the usual “elementary” Selberg sieve with the “analytic” Selberg sieve used in particular in many of the breakthrough works in small gaps between primes; I prefer working with the latter due to its Fourier-analytic flavour.

— 1. Controlling generalised von Mangoldt sums —

To prove (5), we shall first generalise it, by replacing the sequence {\Lambda(n+2)} by a more general sequence {a_n} obeying the following axioms:

  • (i) (Non-negativity) One has {a_n \geq 0} for all {n}.
  • (ii) (Crude size bound) One has {a_n \ll \tau(n)^{O(1)} \log^{O(1)} n} for all {n}, where {\tau} is the divisor function.
  • (iii) (Size) We have {\sum_{n \leq x} a_n = (C+o(1)) x} for some constant {C>0}.
  • (iv) (Elliott-Halberstam type conjecture) For any {\varepsilon,A>0}, one has

    \displaystyle \sum_{d \leq x^{1-\varepsilon}} |\sum_{n \leq x: d|n} a_n - C x \frac{g(d)}{d}| \ll_{\varepsilon,A} x \log^{-A} x

    where {g} is a multiplicative function with {g(p^j) = 1 + O(1/p)} for all primes {p} and {j \geq 1}.

These axioms are a little bit stronger than what is actually needed to make the Bombieri asymptotic sieve work, but we will not attempt to work with the weakest possible axioms here.

We introduce the function

\displaystyle G(s) := \prod_p \frac{1-g(p)/p^s}{1-1/p^s}

which is analytic for {\hbox{Re}(s) > 0}; in particular it can be evaluated at {s=1} to yield

\displaystyle G(1) = \prod_p \frac{1-g(p)/p}{1-1/p}.

There are two model examples of data {a_n, C, g} to keep in mind. The first, discussed in the introduction, is when {a_n =\Lambda(n+2)}, then {C = 2 \Pi_2} and {g} is as in the introduction; one of course needs EH to justify axiom (iv) in this case. The other is when {a_n=1}, in which case {C=1} and {g(n)=1} for all {n}. We will later take advantage of the second example to avoid doing some (routine, but messy) main term computations.

The main result of this section is then

Theorem 1 Let {a_n, g, C, G} be as above. Let {\vec k = (k_1,\dots,k_r)} be a tuple of natural numbers (independent of {x}) that is not equal to {(1,\dots,1)}. Then one has the asymptotic

\displaystyle \sum_{n \leq x} \Lambda_{\vec k}(n) a_n = (G(1)+o(1)) \frac{\prod_{i=1}^r k_i!}{(|\vec k|-1)!} C x \log^{|\vec k|-1} x

as {x \rightarrow \infty}, where {|\vec k| := k_1 + \dots + k_r}.

Note that this recovers (5) (on EH) as a special case.

We now begin the proof of this theorem. Henceforth we allow implied constants in the {O()} or {\ll} notation to depend on {r, \vec k} and {g,G}.

It will be convenient to replace the range {n \leq x} by a shorter range by the following standard localisation trick. Let {B} be a large quantity depending on {r, \vec k} to be chosen later, and let {I} denote the interval {\{ n: x - x \log^{-B} x \leq n \leq x \}}. We will show the estimate

\displaystyle \sum_{n \in I} \Lambda_{\vec k}(n) a_n = (G(1)+o(1)) \frac{\prod_{i=1}^r k_i!}{(|\vec k|-1)!} C |I| \log^{|\vec k|-1} x \ \ \ \ \ (10)

 

from which the original claim follows by a routine summation argument. Observe from axiom (iv) and the triangle inequality that

\displaystyle \sum_{d \leq x^{1-\varepsilon}: \mu^2(d)=1} |\sum_{n \in I: d|n} a_n - C |I| \frac{g(d)}{d}| \ll_{\varepsilon,A} x \log^{-A} x

for any {\varepsilon,A > 0}.

Write {L} for the logarithm function {L(n) := \log n}, thus {\Lambda_k = \mu * L^k} for any {k}. Without loss of generality we may assume that {k_r > 1}; we then factor {\Lambda_{\vec k} = \mu_{\vec k} * L^{k_r}}, where

\displaystyle \mu_{\vec k} := \Lambda_{k_1} * \dots * \Lambda_{k_{r-1}} * \mu.

This function is just {\mu} when {r=1}. When {r>1} the function is more complicated, but we at least have the following crude bound:

Lemma 2 One has the pointwise bound {|\mu_{\vec k}| \leq L^{|\vec k|-k_r}}.

Proof: We induct on {r}. The case {r=1} is obvious, so suppose {r>1} and the claim has already been proven for {r-1}. Since {\mu_{\vec k} = \Lambda_{k_1} * \mu_{(k_2,\dots,k_r)}}, we see from induction hypothesis and the triangle inequality that

\displaystyle |\mu_{\vec k}| \leq \Lambda_{k_1} * L^{|\vec k| - k_r - k_1} \leq L^{|\vec k| - k_r - k_1} (\Lambda_{k_1} * 1).

Since {\Lambda_{k_1}*1 = L^{k_1}} by Möbius inversion, the claim follows. \Box

We can write

\displaystyle \Lambda_{\vec k}(n) = \sum_{d|n} \mu_{\vec k}(d) \log^{k_r} \frac{n}{d}.

In the region {n \in I}, we have {\log^{k_r} \frac{n}{d} = \log^{k_r} \frac{x}{d} + O( \log^{-B+O(1)} x )}. Thus

\displaystyle \Lambda_{\vec k}(n) = \sum_{d|n} \mu_{\vec k}(d) \log^{k_r} \frac{x}{d} + O( \tau(x) \log^{-B+O(1)} x )

for {n \in I}. The contribution of the error term to {O( \tau(x) \log^{-B+O(1)} x )} to (10) is easily seen to be negligible if {B} is large enough, so we may freely replace {\Lambda_{\vec k}(n)} with {\sum_{d|n} \mu_{\vec k}(d) \log^{k_r} \frac{x}{d}} with little difficulty.

If we insert this replacement directly into the left-hand side of (10) and rearrange, we get

\displaystyle \sum_{d \leq x} \mu_{\vec k}(d) \log^{k_r} \frac{x}{d} \sum_{n \in I: d|n} a_d.

We can’t quite control this using axiom (iv) because the range of {d} is a bit too big, as explained in the introduction. So let us introduce a truncated function

\displaystyle \Lambda_{\vec k,\varepsilon}(n) := \sum_{d|n} \mu_{\vec k}(d) \log^{k_r} \frac{x}{d} \eta_\varepsilon( \frac{\log d}{\log x} ) \ \ \ \ \ (11)

 

where {\varepsilon>0} is a small quantity to be chosen later, and {\eta_\varepsilon: {\bf R} \rightarrow [0,1]} is a smooth function that equals {1} on {(-\infty,1-4\varepsilon)} and equals {0} on {(1-3\varepsilon,+\infty)}. Suppose one could establish the following two estimates for any fixed {\varepsilon>0}:

\displaystyle \sum_{n \in I} \Lambda_{\vec k}(n) a_n = \sum_{n \in I} \Lambda_{\vec k,\varepsilon}(n) a_n + O( (\varepsilon+o(1)) C |I| \log^{|\vec k|-1} x ) \ \ \ \ \ (12)

 

and

\displaystyle \sum_{n \in I} \Lambda_{\vec k,\varepsilon}(n) a_n = C Q_{\varepsilon,x} G(1) + o( |I| \log^{|\vec k|-1} x ) \ \ \ \ \ (13)

 

where {Q_{\varepsilon,x}} is a quantity that depends on {\varepsilon, \eta_\varepsilon, \vec k, B, x} but not on {C, g,G}. Then on combining the two estimates we would have

\displaystyle \sum_{n \in I} \Lambda_{\vec k}(n) a_n = C Q_{\varepsilon,x} G(1) + (O(\varepsilon) + o(1)) C |I| \log^{|\vec k|-1} x. \ \ \ \ \ (14)

 

One could in principle compute {Q_{\varepsilon,x}} explicitly from the proof of (13), but one can avoid doing so by the following comparison trick. In the special case {a_n=1}, standard multiplicative number theory (noting that the Dirichlet series {\sum_n \frac{\Lambda_{\vec k}(n)}{n^s}} has a pole of order {|\vec k|} at {s=1}, with top Laurent coefficient {\prod_{j=1}^r k_j!}) gives the asymptotic

\displaystyle \sum_{n \in I} \Lambda_{\vec k}(n) a_n = \frac{\prod_{i=1}^r k_i!}{(|\vec k|-1)!} + o(1)) |I| \log^{|\vec k|-1} x

which when compared with (14) for {a_n=1} (recalling that {G(1)=C=1} in this case) gives the formula

\displaystyle Q_{\varepsilon,x} = (\prod_{j=1}^r k_j + O(\varepsilon)) |I| \log^{|\vec k|-1} x.

Inserting this back into (14) and recalling that {\varepsilon>0} can be made arbitrarily small, we obtain (10).

As it turns out, the estimate (13) is easy to establish, but the estimate (12) is not, roughly speaking because the typical number {n} in {I} has too many divisors {d} in the range {[x^{1-4\varepsilon},1]}, each of which gives a contribution to the error term. (In the book of Friedlander and Iwaniec, the estimate (13) is established anyway, but only after assuming a stronger version of (iv), roughly speaking in which {d} is allowed to be as large as {x \exp( -\log^{1/4} x)}.) To resolve this issue, we will insert a preliminary sieve {\nu_\varepsilon} that will remove most of the potential divisors {d} i the range {[x^{1-4\varepsilon},1]} (leaving only about {O(1)} such divisors on the average for typical {n}), making the analogue of (12) easier to prove (at the cost of making the analogue of (13) more difficult). Namely, if one can find a function {\nu_\varepsilon: {\bf N} \rightarrow {\bf R}} for which one has the estimates

\displaystyle \sum_{n \in I} \Lambda_{\vec k}(n) a_n = \sum_{n \in I} \Lambda_{\vec k}(n) \nu_\varepsilon(n) a_n + O( (\varepsilon+o(1)) C |I| \log^{|\vec k|-1} x ), \ \ \ \ \ (15)

 

\displaystyle \sum_{n \in I} \Lambda_{\vec k}(n) \nu_\varepsilon(n) a_n

\displaystyle = \sum_{n \in I} \Lambda_{\vec k,\varepsilon}(n) \nu_\varepsilon(n) a_n + O( (\varepsilon+o(1)) C |I| \log^{|\vec k|-1} x ) \ \ \ \ \ (16)

 

and

\displaystyle \sum_{n \in I} \Lambda_{\vec k,\varepsilon}(n) \nu_\varepsilon(n) a_n = C Q'_{\varepsilon,x} G(1) + o( |I| \log^{|\vec k|-1} x ) \ \ \ \ \ (17)

 

for some quantity {Q'_{\varepsilon,x}} that depends on {\varepsilon, \eta_\varepsilon, \vec k, B, x} but not on {C, g, G,}, then by repeating the previous arguments we will again be able to establish (10).

The key estimate is (16). As we shall see, when comparing {\Lambda_{\vec k}(n) \nu_\varepsilon(n)} with {\Lambda_{\vec k,\varepsilon}(n) \nu_\varepsilon(n)}, the weight {\nu_\varepsilon} will cost us a factor of {1/\varepsilon}, but the {\log^{k_r} \frac{x}{d}} term in the definitions of {\Lambda_{\vec k}} and {\Lambda_{\vec k,\varepsilon}} will recover a factor of {\varepsilon^{k_r}}, which will give the desired bound since we are assuming {k_r > 1}.

One has some flexibility in how to select the weight {\nu_\varepsilon}: basically any standard sieve that uses divisors of size at most {x^{2\varepsilon}} to localise (at least approximately) to numbers that are rough in the sense that they have no (or at least very few) factors less than {x^\varepsilon}, will do. We will use the analytic Selberg sieve choice

\displaystyle \nu_\varepsilon(n) := (\sum_{d|n} \mu(d) \psi( \frac{\log d}{\varepsilon \log x} ))^2 \ \ \ \ \ (18)

 

where {\psi: {\bf R} \rightarrow [0,1]} is a smooth function supported on {[-1,1]} that equals {1} on {[-1/2,1/2]}.

It remains to establish the bounds (15), (16), (17). To warm up and introduce the various methods needed, we begin with the standard bound

\displaystyle \sum_{n \in I} \nu_\varepsilon(n) a_n = \frac{C|I|}{\varepsilon \log x} (\int_0^1 \psi'(u)^2\ du) G(1) + o(1)), \ \ \ \ \ (19)

 

where {\psi'} denotes the derivative of {\psi}. Note the loss of {1/\varepsilon} that had previously been pointed out. In the arguments that follows I will be a little brief with the details, as they are standard (see e.g. this previous post).

We now prove (19). The left-hand side can be expanded as

\displaystyle \sum_{d_1,d_2} \mu(d_1) \mu(d_2) \psi( \frac{\log d_1}{\varepsilon \log x} ) \psi( \frac{\log d_2}{\varepsilon \log x} ) \sum_{n \in I: [d_1,d_2]|n} a_n

where {[d_1,d_2]} denotes the least common multiple of {d_1} and {d_2}. From the support of {\psi} we see that the summand is only non-vanishing when {[d_1,d_2] \leq x^{2\varepsilon}}. We now use axiom (iv) and split the left-hand side into a main term

\displaystyle \sum_{d_1,d_2} \mu(d_1) \mu(d_2) \psi( \frac{\log d_1}{\varepsilon \log x} ) \psi( \frac{\log d_2}{\varepsilon \log x} ) \frac{g(d)}{d} C |I|

and an error term that is at most

\displaystyle O_\varepsilon( \sum_{d \leq x^{2\varepsilon}} \tau(d)^{O(1)} | \sum_{n \in I: d|n} a_n - \frac{g(d)}{d} C |I|| ). \ \ \ \ \ (20)

 

From axiom (ii) and elementary multiplicative number theory, we have the bound

\displaystyle \sum_{d \leq x} \tau(d)^{O(1)} | \sum_{n \in I: d|n} a_n - \frac{g(d)}{d} C |I| \ll C |I| \log^{O(1)} x

so from axiom (iv) and Cauchy-Schwarz we see that the error term (20) is acceptable. Thus it will suffice to establish the bound

\displaystyle \sum_{d_1,d_2} \mu(d_1) \mu(d_2) \psi( \frac{\log d_1}{\varepsilon \log x} ) \psi( \frac{\log d_2}{\varepsilon \log x} ) \frac{g([d_1,d_2])}{[d_1,d_2]}

\displaystyle = \frac{1}{\varepsilon \log x} (\int_0^1 \psi'(u)^2\ du) G(1) + o(\frac{1}{\log x}). \ \ \ \ \ (21)

 

The summand here is almost, but not quite, multiplicative in {d_1,d_2}. To make it genuinely multiplicative, we perform a (shifted) Fourier expansion

\displaystyle \psi(u) = \int_{\bf R} e^{-(1+it)u} \Psi(t)\ dt \ \ \ \ \ (22)

 

for some rapidly decreasing function {\Psi} (essentially the Fourier transform of {e^u \psi(u)}). Thus

\displaystyle \psi( \frac{\log d}{\varepsilon \log x} ) = \int_{\bf R} \frac{1}{d^{\frac{1+it}{\varepsilon \log x}}} \Psi(t)\ dt,

and so the left-hand side of (21) can be rearranged using Fubini’s theorem as

\displaystyle \int_{\bf R} \int_{\bf R} E(\frac{1+it_1}{\varepsilon \log x},\frac{1+it_2}{\varepsilon \log x})\ \Psi(t_1) \Psi(t_2) dt_1 dt_2 \ \ \ \ \ (23)

 

where

\displaystyle E(s_1,s_2) := \sum_{d_1,d_2} \frac{\mu(d_1) \mu(d_2)}{d_1^{s_1}d_2^{s_2}} \frac{g([d_1,d_2])}{[d_1,d_2]}.

We can factorise {E(s_1,s_2)} as an Euler product:

\displaystyle E(s_1,s_2) = \prod_p (1 - \frac{g(p)}{p^{1+s_1}} - \frac{g(p)}{p^{1+s_2}} + \frac{g(p)}{p^{1+s_1+s_2}}).

Taking absolute values and using Mertens’ theorem leads to the crude bound

\displaystyle E(\frac{1+it_1}{\varepsilon \log x},\frac{1+it_2}{\varepsilon \log x}) \ll_\varepsilon \log^{O(1)} x

which when combined with the rapid decrease of {\Psi}, allows us to restrict the region of integration in (23) to the square {\{ |t_1|, |t_2| \leq \sqrt{\log x} \}} (say) with negligible error. Next, we use the Euler product

\displaystyle \zeta(s) = \prod_p (1-\frac{1}{p^s})^{-1}

for {\hbox{Re} s > 1} to factorise

\displaystyle E(s_1,s_2) = \frac{\zeta(1+s_1+s_2)}{\zeta(1+s_1) \zeta(1+s_2)} \prod_p E_p(s_1,s_2)

where

\displaystyle E_p(s_1,s_2) := \frac{(1 - \frac{g(p)}{p^{1+s_1}} - \frac{g(p)}{p^{1+s_2}} + \frac{g(p)}{p^{1+s_1+s_2}})(1 - \frac{1}{p^{1+s_1+s_2}})}{(1-\frac{1}{p^{1+s_1}})(1-\frac{1}{p^{1+s_2}})}.

For {s_1,s_2=o(1)} with nonnegative real part, one has

\displaystyle E_p(s_1,s_2) = 1 + O(1/p^2)

and so by the Weierstrass {M}-test, {\prod_p E_p(s_1,s_2)} is continuous at {s_1=s_2=0}. Since

\displaystyle \prod_p E_p(0,0) = G(1)

we thus have

\displaystyle \prod_p E_p(s_1,s_2) = G(1) + o(1)

Also, since {\zeta} has a pole of order {1} at {s=1} with residue {1}, we have

\displaystyle \frac{\zeta(1+s_1+s_2)}{\zeta(1+s_1) \zeta(1+s_2)} = (1+o(1)) \frac{s_1 s_2}{s_1+s_2}

and thus

\displaystyle E(s_1,s_2) = (G(1)+o(1)) \frac{s_1s_2}{s_1+s_2}.

The quantity (23) can thus be written, up to errors of {o(\frac{1}{\log x})}, as

\displaystyle \frac{G(1)}{\varepsilon \log x} \int_{|t_1|, |t_2| \leq \sqrt{\log x}} \frac{(1+it_1)(1+it_2)}{1+it_1+1+it_2} \Psi(t_1) \Psi(t_2)\ dt_1 dt_2.

Using the rapid decrease of {\Psi}, we may remove the restriction on {t_1,t_2}, and it will now suffice to prove the identity

\displaystyle \int_{\bf R} \int_{\bf R} \frac{(1+it_1)(1+it_2)}{1+it_1+1+it_2} \Psi(t_1) \Psi(t_2)\ dt_1 dt_2 = (\int_0^1 \psi'(u)^2\ du)^2.

But on differentiating and then squaring (22) we have

\displaystyle \psi'(u)^2 = \int_{\bf R} \int_{\bf R} (1+it_1)(1+it_2) e^{-(1+it_1+1+it_2)u}\Psi(t_1) \Psi(t_2)\ dt_1 dt_2

and the claim follows by integrating in {u} from zero to infinity (noting that {\psi'} vanishes for {u>1}).

We have the following variant of (19):

Lemma 3 For any {d \leq x^{1-3\varepsilon}}, one has

\displaystyle \sum_{n \in I: d|n} \nu_\varepsilon(n) a_n \ll \frac{C|I|}{\varepsilon \log x} \frac{\prod_{p|d} O( \min( \frac{\log p}{\varepsilon \log x}, 1 )^2 )}{d} + R_d \ \ \ \ \ (24)

 

where the {R_d} are such that

\displaystyle \sum_{d \leq x^{1-3\varepsilon}} R_d \ll_A |I| \log^{-A} x \ \ \ \ \ (25)

 

for any {A>0}. We also have the variant

\displaystyle \sum_{n \in I: d|n} \nu_\varepsilon(n/d) a_n \ll \frac{C|I|}{\varepsilon \log x} \frac{\prod_{p|d} O(1 ) )}{d} + R_d. \ \ \ \ \ (26)

 

If in addition {d} has no prime factors less than {x^\delta} for some fixed {\delta>0}, one has

\displaystyle \sum_{n \in I: d|n} \nu_\varepsilon(n) a_n

\displaystyle = \frac{1+o(1)}{d} \frac{C|I|}{\varepsilon \log x} (\int_0^1 \psi'(u)^2\ du) G(1) + O(R_d). \ \ \ \ \ (27)

 

Roughly speaking, the above estimates assert that {\nu_\varepsilon} is concentrated on those numbers {n} with no prime factors much less than {x^\varepsilon}, but factors {d} without such small prime divisors occur with about the same relative density as they do in the integers.

Proof: The left-hand side of (24) can be expanded as

\displaystyle \sum_{d_1,d_2} \mu(d_1) \mu(d_2) \psi( \frac{\log d_1}{\varepsilon \log x} ) \psi( \frac{\log d_2}{\varepsilon \log x} ) \sum_{n \in I: [d_1,d_2,d]|n} a_n.

If we define

\displaystyle R_d := \sum_{d' \leq x^{1-\varepsilon}: d|d'} \tau(d')^2 |\sum_{n \in I:d'|n} a_n - \frac{g(d')}{d'} C|I||

then the previous expression can be written as

\displaystyle \sum_{d_1,d_2} \mu(d_1) \mu(d_2) \psi( \frac{\log d_1}{\varepsilon \log x} ) \psi( \frac{\log d_2}{\varepsilon \log x} ) \frac{g([d_1,d_2,d])}{[d_1,d_2,d]} C|I| + O(R_d),

while one has

\displaystyle \sum_{d \leq x^{1-3\varepsilon}} R_d \leq \sum_{d' \leq x^{1-\varepsilon}} \tau(d')^3 |\sum_{n \in I:d'|n} a_n - \frac{g(d')}{d'} C|I||

which gives (25) from Axiom (iv). To prove (24), it now suffices to show that

\displaystyle \sum_{d_1,d_2} \mu(d_1) \mu(d_2) \psi( \frac{\log d_1}{\varepsilon \log x} ) \psi( \frac{\log d_2}{\varepsilon \log x} ) \frac{g([d_1,d_2,d])}{[d_1,d_2,d]}

\displaystyle \ll \frac{1}{\varepsilon \log x} \frac{\prod_{p|d} O( \min( \frac{\log p}{\varepsilon \log x}, 1 )^2 )}{d}. \ \ \ \ \ (28)

 

Arguing as before, the left-hand side is

\displaystyle \int_{\bf R} \int_{\bf R} E^{(d)}(\frac{1+it_1}{\varepsilon \log x},\frac{1+it_2}{\varepsilon \log x})\ \Psi(t_1) \Psi(t_2) dt_1 dt_2

where

\displaystyle E^{(d)}(s_1,s_2) := \sum_{d_1,d_2} \frac{\mu(d_1) \mu(d_2)}{d_1^{s_1}d_2^{s_2}} \frac{g([d_1,d_2,d])}{[d_1,d_2,d]}.

From Mertens’ theorem we have

\displaystyle E^{(d)}(s_1,s_2) \ll_\varepsilon \frac{\prod_{p|d} O(1)}{d} \log^{O(1)} x

when {\hbox{Re} s_1, \hbox{Re} s_2 = \frac{1}{\varepsilon \log x}}, so the contribution of the terms where {|t_1|, |t_2| \geq \sqrt{\log x}} can be absorbed into the {R_d} error (after increasing that error slightly). For the remaining contributions, we see that

\displaystyle E^{(d)}(s_1,s_2) = \frac{\zeta(1+s_1+s_2)}{\zeta(1+s_1) \zeta(1+s_2)} \prod_p E^{(d)}_p(s_1,s_2)

where {E^{(d)}_p(s_1,s_2) = E_p(s_1,s_2)} if {p} does not divide {d}, and

\displaystyle E^{(d)}_p(s_1,s_2) = \frac{g(p^j)}{p^j} \frac{(1 - \frac{1}{p^{s_1}}) (1 - \frac{1}{p^{s_2}}) (1 - \frac{1}{p^{1+s_1+s_2}})}{(1-\frac{1}{p^{1+s_1}})(1-\frac{1}{p^{1+s_2}})}

if {p} divides {d} {j} times for some {j \geq 1}. In the latter case, Taylor expansion gives the bounds

\displaystyle |E^{(d)}_p(\frac{1+it_1}{\varepsilon \log x},\frac{1+it_2}{\varepsilon \log x})| \lesssim (1+|t_1|+|t_2|)^{O(1)} \frac{\min( \frac{\log p}{\varepsilon \log x}, 1 )^2}{p}

and the claim (28) follows. When {p \geq x^\delta} and {|t_1|, |t_2| \leq \sqrt{\log x}} we have

\displaystyle E^{(d)}_p(\frac{1+it_1}{\varepsilon \log x},\frac{1+it_2}{\varepsilon \log x}) = \frac{1+o(1)}{p^j}

and (27) follows by repeating the previous calculations. Finally, (26) is proven similarly to (24) (using {d[d_1,d_2]} in place of {[d_1,d_2,d]}). \Box

Now we can prove (15), (16), (17). We begin with (15). Using the Leibniz rule {L(f*g) = (Lf)*g + f*(Lg)} applied to the identity {\mu = \mu * 1 * \mu} and using {\Lambda = \mu*L} and Möbius inversion (and the associativity and commutativity of Dirichlet convolution) we see that

\displaystyle L\mu = - \mu * \Lambda. \ \ \ \ \ (29)

 

Next, by applying the Leibniz rule to {\Lambda_k = \mu * L^k} for some {k \geq 1} and using (29) we see that

\displaystyle L \Lambda_k = L \mu * L^k + \mu * L^{k+1}

\displaystyle = - \mu * \Lambda * L^k + \Lambda_{k+1}

and hence we have the recursive identity

\displaystyle \Lambda_{k+1} = L \Lambda_k + \Lambda *\Lambda_k. \ \ \ \ \ (30)

 

In particular, from induction we see that {\Lambda_k} is supported on numbers with at most {k} distinct prime factors, and hence {\Lambda_{\vec k}} is supported on numbers with at most {|\vec k|} distinct prime factors. In particular, from (18) we see that {\nu_\varepsilon(n) = O(1)} on the support of {\Lambda_{\vec k}}. Thus it will suffice to show that

\displaystyle \sum_{n \in I: \nu_\varepsilon(n) \neq 1} \Lambda_{\vec k}(n) a_n \ll (\varepsilon+o(1)) C |I| \log^{|\vec k|-1} x.

If {\nu_\varepsilon(n) \neq 1} and {\Lambda_{\vec k}(n) \neq 0}, then {n} has at most {|\vec k|} distinct prime factors {p_1 < p_2 < \dots < p_r}, with {p_1 \leq x^\varepsilon}. If we factor {n = n_1 n_2}, where {n_1} is the contribution of those {p_i} with {p_i \leq x^{1/10|\vec k|}}, and {n_2} is the contribution of those {p_i} with {p_i > x^{1/10|\vec k|}}, then at least one of the following two statements hold:

  • (a) {n_1} (and hence {n}) is divisible by a square number of size at least {x^{1/10}}.
  • (b) {n_1 \leq x^{1/5}}.

The contribution of case (a) is easily seen to be acceptable by axiom (ii). For case (b), we observe from (30) and induction that

\displaystyle \Lambda_k(n) \ll \log^{|\vec k|} x \prod_{j=1}^k \frac{\log p_j}{\log x}

and so it will suffice to show that

\displaystyle \sum_{n_1} (\prod_{p|n_1} \frac{\log p}{\log x}) \sum_{n \in I: n_1 | n} 1_R(n/n_1) a_n \ll (\varepsilon + o(1)) C |I| \log^{-1} x

where {n_1} ranges over numbers bounded by {x^{1/5}} with at most {|\vec k|} distinct prime factors, the smallest of which is at most {x^\varepsilon}, and {R} consists of those numbers with no prime factor less than or equal to {x^{1/10|\vec k|}}. Applying (26) (with {\varepsilon} replaced by {1/10|\vec k|}) gives the bound

\displaystyle \sum_{n \in I: d|n} 1_R(n/n_1) a_n \ll \frac{C|I|}{\log x} \frac{1}{n_1} + R_d

so by (25) it suffices to show that

\displaystyle \sum_{n_1} (\prod_{p|n_1} \frac{\log p}{\log x}) \frac{1}{n_1} \ll \varepsilon

subject to the same constraints on {n_1} as before. The contribution of those {n_1} with {r} distinct prime factors can be bounded by

\displaystyle O(\sum_{p_1 \leq x^\varepsilon} \frac{\log p_1}{p_1 \log x}) \times O(\sum_{p \leq x^{1/5}} \frac{\log p}{p\log x})^{r-1};

applying Mertens’ theorem and summing over {1 \leq r \leq |\vec k|}, one obtains the claim.

Now we show (16). As discussed previously in this section, we can replace {\Lambda_{\vec k}(n)} by {\sum_{d|n} \mu_{\vec k}(d) \log^{k_r} \frac{x}{d}} with negligible error. Comparing this with (16) and (11), we see that it suffices to show that

\displaystyle \sum_{n \in I} \sum_{d|n} \mu_{\vec k}(d) \log^{k_r} \frac{x}{d} (1 - \eta_\varepsilon(\frac{\log d}{\log x})) \nu_\varepsilon(n) a_n \ll (\varepsilon+o(1)) C |I| \log^{|\vec k|-1} x.

From the support of {\eta_\varepsilon}, the summand on the left-hand side is only non-zero when {d \geq x^{1-4\varepsilon}}, which makes {\log^{k_r} \frac{x}{d} \ll \varepsilon^{k_r} \log^{k_r} x \leq \varepsilon^2 \log^{k_r} x}, where we use the crucial hypothesis {k_r > 1} to gain enough powers of {\varepsilon} to make the argument here work. Applying Lemma 2, we reduce to showing that

\displaystyle \sum_{n \in I} \sum_{d|n: d \geq x^{1-4\varepsilon}} \nu_\varepsilon(n) a_n \ll \frac{1+o(1)}{\varepsilon \log x} C |I|.

We can make the change of variables {d \mapsto n/d} to flip the sum

\displaystyle \sum_{d|n: d \geq x^{1-4\varepsilon}} 1 \leq \sum_{d|n: d \leq x^{3\varepsilon}} 1

and then swap the sums to reduce to showing that

\displaystyle \sum_{d \leq x^{4\varepsilon}} \sum_{n \in I} \nu_\varepsilon(n) a_n \ll \frac{1+o(1)}{\varepsilon \log x} C |I|.

By Lemma 3, it suffices to show that

\displaystyle \sum_{d \leq x^{4\varepsilon}} \frac{\prod_{p|d} O( \min( \frac{\log p}{\varepsilon \log x}, 1 )^2 )}{d} \ll 1.

To prove this, we use the Rankin trick, bounding the implied weight {1_{d \leq x^{4\varepsilon}}} by {O( \frac{1}{d^{1/\varepsilon \log x}} )}. We can then bound the left-hand side by the Euler product

\displaystyle \prod_p (1 + O( \frac{\min( \frac{\log p}{\varepsilon \log x}, 1 )^2}{p^{1+1/\varepsilon \log x}} ))

which can be bounded by

\displaystyle \exp( O( \sum_p \frac{\min( \frac{\log p}{\varepsilon \log x}, 1 )^2}{p^{1+1/\varepsilon \log x}} ) )

and the claim follows from Mertens’ theorem.

Finally, we show (17). By (11), the left-hand side expands as

\displaystyle \sum_{d \leq x^{1-3\varepsilon}} \mu_{\vec k}(d) \log^{k_r} \frac{x}{d} \eta_\varepsilon(\frac{\log d}{\log x}) \sum_{n \in I: d|n} \nu_\varepsilon(n) a_n.

We let {\delta>0} be a small constant to be chosen later. We divide the outer sum into two ranges, depending on whether {d} only has prime factors greater than {x^\delta} or not. In the former case, we can apply (27) to write this contribution as

\displaystyle \sum_{d \leq x^{1-3\varepsilon}} \mu_{\vec k}(d) \log^{k_r} \frac{x}{d} \eta_\varepsilon(\frac{\log d}{\log x}) \frac{1+o(1)}{d} \frac{C|I|}{\varepsilon \log x} (\int_0^1 \psi'(u)^2\ du) G(1)

plus a negligible error, where the {d} is implicitly restricted to numbers with all prime factors greater than {x^\delta}. The main term is messy, but it is of the required form {C Q'_{\varepsilon,x} G(1)} up to an acceptable error, so there is no need to compute it any further. It remains to consider those {d} that have at least one prime factor less than {x^\delta}. Here we use (24) instead of (27) as well as Lemma 3 to dominate this contribution by

\displaystyle \sum_{d \leq x^{1-3\varepsilon}} O( \log^{|\vec k|} x \frac{C|I|}{\varepsilon \log x} \frac{\prod_{p|d} O( \min( \frac{\log p}{\varepsilon \log x}, 1 )^2 )}{d} )

up to negligible errors, where {d} is now restricted to have at least one prime factor less than {x^\delta}. This makes at least one of the factors {\min( \frac{\log p}{\varepsilon \log x}, 1 )} to be at most {O_\varepsilon(\delta)}. A routine application of Rankin’s trick shows that

\displaystyle \sum_{d \leq x^{1-3\varepsilon}} \frac{\prod_{p|d} O( \min( \frac{\log p}{\varepsilon \log x}, 1 ) )}{d} \ll_\varepsilon 1

and so the total contribution of this case is {O_\varepsilon((\delta+o(1)) |I| \log^{|\vec k|-1} x)}. Since {\delta>0} can be made arbitrarily small, (17) follows.

— 2. Weierstrass approximation —

Having proved Theorem 1, we now take linear combinations of this theorem, combined with the Weierstrass approximation theorem, to give the asymptotics (7), (8) described in the introduction.

Let {a_n}, {g}, {C}, {G} be as in that theorem. It will be convenient to normalise the weights {\Lambda_{\vec k}} by {L^{1-|\vec k|}} to make their mean value comparable to {1}. From Theorem 1 and summation by parts we have

\displaystyle \sum_{n \leq x} L^{1-|\vec k|} \Lambda_{\vec k}(n) a_n = (G(1)+o(1)) \frac{\prod_{i=1}^r k_i!}{(|\vec k|-1)!} C x \ \ \ \ \ (31)

 

whenever {\vec k} does not consist entirely of ones.

We now take a closer look at what happens when {\vec k} does consist entirely of ones. Let {1^r} denote the {r}-tuple {(1,\dots,1)}. Convolving the {k=1} case of (30) with {r-1} copies of {\Lambda} for some {r \geq 1} and using the Leibniz rule, we see that

\displaystyle \Lambda_{(1^{r-1}, 2)} = \frac{1}{r} L \Lambda_{1^r} + \Lambda_{1^{r+1}}

and hence

\displaystyle L^{-r} \Lambda_{1^{r+1}} = L^{-r} \Lambda_{(1^{r-1},2)} - \frac{1}{r} L^{1-r} \Lambda_{1^r}.

Multiplying by {a_n} and summing over {n \leq x}, and using (31) to control the {\Lambda_{(1^{r-1},2)}} term, one has

\displaystyle \sum_{n \leq x} L^{-r} \Lambda_{1^{r+1}}(n) a_n = (G(1)+o(1)) \frac{2}{r!} - \frac{1}{r} \sum_{n \leq x} L^{1-r} \Lambda_{1^{r}}(n) a_n.

If we define {\delta_x} (up to an error of {o(1)}) by the formula

\displaystyle \sum_{n \leq x} \Lambda(n) a_n = (\delta_x G(1) + o(1)) C x

then an induction then shows that

\displaystyle \sum_{n \leq x} L^{1-r} \Lambda_{1^r}(n) a_n = \frac{1}{(r-1)!} (\delta_x G(1) + o(1)) C x

for odd {r}, and

\displaystyle \sum_{n \leq x} L^{1-r} \Lambda_{1^r}(n) a_n = \frac{1}{(r-1)!} ((2-\delta_x) G(1) + o(1)) C x

for even {r}. In particular, after adjusting {\delta_x} by {o(1)} if necessary, we have {0 \leq \delta_x \leq 2} since the left-hand sides are non-negative.

If we now define the comparison sequence {b_n := C G(1) (1 + (1-\delta_x) \mu(n))}, standard multiplicative number theory shows that the above estimates also hold when {a_n} is replaced by {b_n}; thus

\displaystyle \sum_{n \leq x} L^{1-r} \Lambda_{1^r}(n) a_n = \sum_{n \leq x} L^{1-r} \Lambda_{1^r}(n) b_n + o( x )

for both odd and even {r}. The bound (31) also holds for {b_n} when {\vec k} does not consist entirely of ones, and hence

\displaystyle \sum_{n \leq x} L^{1-|\vec k|} \Lambda_{\vec k}(n) a_n = \sum_{n \leq x} L^{1-|\vec k|} \Lambda_{\vec k}(n) b_n + o( x )

for any fixed {\vec k} (which may or may not consist entirely of ones).

Next, from induction (on {j_1+\dots+j_r}), the Leibniz rule, and (30), we see that for any {r \geq 1} and {j_1,\dots,j_r \geq 0}, {k_1,\dots,k_r}, the function

\displaystyle L^{1-j_1-\dots-j_r-|\vec k|} ((L^{j_1} \Lambda_{k_1}) * \dots * (L^{j_r} \Lambda_{k_r})) \ \ \ \ \ (32)

 

is a finite linear combination of functions of the form {L^{1-|\vec k'|} \Lambda_{\vec k'}} for tuples {\vec k'} that may possibly consist entirely of ones. We thus have

\displaystyle \sum_{n \leq x} f(n) a_n = \sum_{n \leq x}f(n) b_n + o( x )

whenever {f} is one of these functions (32). Specialising to the case {k_1=\dots=k_r=1}, we thus have

\displaystyle \sum_{n_1 \dots n_r \leq x} a_{n} \log^{1-r} n \prod_{i=1}^r (\log n_i/\log n)^{j_i} \Lambda(n_i)

\displaystyle = \sum_{n_1 \dots n_r \leq x} b_{n} \log^{1-r} n \prod_{i=1}^r (\log n_i/\log n)^{j_i} \Lambda(n_i) + o(x )

where {n := n_1 \dots n_r}. The contribution of those {n_i} that are powers of primes can be easily seen to be negligible, leading to

\displaystyle \sum_{p_1 \dots p_r \leq x} a_{n} \log n \prod_{i=1}^r (\log p_i/\log n)^{j_i+1}

\displaystyle = \sum_{p_1 \dots p_r \leq x} b_{n} \prod_{i=1}^r (\log p_i/\log n)^{j_i+1} + o(x)

where now {n := p_1 \dots p_r}. The contribution of the case where two of the primes {p_i} agree can also be seen to be negligible, as can the error when replacing {\log n} with {\log x}, and then by symmetry

\displaystyle \sum_{p_1 \dots p_r \leq x: p_1 < \dots < p_r} a_{n} \prod_{i=1}^r (\log p_i/\log n)^{j_i+1}

\displaystyle = \sum_{p_1 \dots p_r \leq x: p_1 < \dots < p_r} b_{n} \prod_{i=1}^r (\log p_i/\log n)^{j_i+1} + o(x / \log x).

By linearity, this implies that

\displaystyle \sum_{p_1 \dots p_r \leq x: p_1 < \dots < p_r} a_{n} P( \log p_1/\log n, \dots, \log p_r/\log n)

\displaystyle = \sum_{p_1 \dots p_r \leq x: p_1 < \dots < p_r} b_{n} P( \log p_1/\log n, \dots, \log p_r/\log n) + o(x / \log x)

for any polynomial {P(t_1,\dots,t_r)} that vanishes on the coordinate hyperplanes {t_i=0}. The right-hand side can also be evaluated by Mertens’ theorem as

\displaystyle CG(1) \delta_x \int_{\Delta_r} P x + o(x)

when {r} is odd and

\displaystyle CG(1) (2-\delta_x) \int_{\Delta_r} P x + o(x)

when {r} is even. Using the Weierstrass approximation theorem, we then have

\displaystyle \sum_{p_1 \dots p_r \leq x: p_1 < \dots < p_r} a_{n} g_r( \log p_1/\log n, \dots, \log p_r/\log n)

\displaystyle = \sum_{p_1 \dots p_r \leq x: p_1 < \dots < p_r} b_{n} g_r( \log p_1/\log n, \dots, \log p_r/\log n) + o(x / \log x)

for any continuous function {g_r} that is compactly supported in the interior of {\Delta_r}. Computing the right-hand side using Mertens’ theorem as before, we obtain the claimed asymptotics (7), (8).

Remark 4 The Bombieri asymptotic sieve has to use the full power of EH (or GEH); there are constructions due to Ford that show that if one only has a distributional hypothesis up to {x^{1-c}} for some fixed constant {c>0}, then the asymptotics of sums such as (5), or more generally (9), are not determined by a single scalar parameter {\delta_x}, but can also vary in other ways as well. Thus the Bombieri asymptotic sieve really is asymptotic; in order to get {o(1)} type error terms one needs the level {1-\varepsilon} of distribution to be asymptotically equal to {1} as {x \rightarrow \infty}. Related to this, the quantitative decay of the {o(1)} error terms in the Bombieri asymptotic sieve are extremely poor; in particular, they depend on the dependence of implied constant in axiom (iv) on the parameters {\varepsilon,A}, for which there is no consensus on what one should conjecturally expect.

I’ve just uploaded to the arXiv my paper “Equivalence of the logarithmically averaged Chowla and Sarnak conjectures“, submitted to the Festschrift “Number Theory – Diophantine problems, uniform distribution and applications” in honour of Robert F. Tichy. This paper is a spinoff of my previous paper establishing a logarithmically averaged version of the Chowla (and Elliott) conjectures in the two-point case. In that paper, the estimate

\displaystyle  \sum_{n \leq x} \frac{\lambda(n) \lambda(n+h)}{n} = o( \log x )

as {x \rightarrow \infty} was demonstrated, where {h} was any positive integer and {\lambda} denoted the Liouville function. The proof proceeded using a method I call the “entropy decrement argument”, which ultimately reduced matters to establishing a bound of the form

\displaystyle  \sum_{n \leq x} \frac{|\sum_{h \leq H} \lambda(n+h) e( \alpha h)|}{n} = o( H \log x )

whenever {H} was a slowly growing function of {x}. This was in turn established in a previous paper of Matomaki, Radziwill, and myself, using the recent breakthrough of Matomaki and Radziwill.

It is natural to see to what extent the arguments can be adapted to attack the higher-point cases of the logarithmically averaged Chowla conjecture (ignoring for this post the more general Elliott conjecture for other bounded multiplicative functions than the Liouville function). That is to say, one would like to prove that

\displaystyle  \sum_{n \leq x} \frac{\lambda(n+h_1) \dots \lambda(n+h_k)}{n} = o( \log x )

as {x \rightarrow \infty} for any fixed distinct integers {h_1,\dots,h_k}. As it turns out (and as is detailed in the current paper), the entropy decrement argument extends to this setting (after using some known facts about linear equations in primes), and allows one to reduce the above estimate to an estimate of the form

\displaystyle  \sum_{n \leq x} \frac{1}{n} \| \lambda \|_{U^d[n, n+H]} = o( \log x )

for {H} a slowly growing function of {x} and some fixed {d} (in fact we can take {d=k-1} for {k \geq 3}), where {U^d} is the (normalised) local Gowers uniformity norm. (In the case {k=3}, {d=2}, this becomes the Fourier-uniformity conjecture discussed in this previous post.) If one then applied the (now proven) inverse conjecture for the Gowers norms, this estimate is in turn equivalent to the more complicated looking assertion

\displaystyle  \sum_{n \leq x} \frac{1}{n} \sup |\sum_{h \leq H} \lambda(n+h) F( g^h x )| = o( \log x ) \ \ \ \ \ (1)

where the supremum is over all possible choices of nilsequences {h \mapsto F(g^h x)} of controlled step and complexity (see the paper for definitions of these terms).

The main novelty in the paper (elaborating upon a previous comment I had made on this blog) is to observe that this latter estimate in turn follows from the logarithmically averaged form of Sarnak’s conjecture (discussed in this previous post), namely that

\displaystyle  \sum_{n \leq x} \frac{1}{n} \lambda(n) F( T^n x )= o( \log x )

whenever {n \mapsto F(T^n x)} is a zero entropy (i.e. deterministic) sequence. Morally speaking, this follows from the well-known fact that nilsequences have zero entropy, but the presence of the supremum in (1) means that we need a little bit more; roughly speaking, we need the class of nilsequences of a given step and complexity to have “uniformly zero entropy” in some sense.

On the other hand, it was already known (see previous post) that the Chowla conjecture implied the Sarnak conjecture, and similarly for the logarithmically averaged form of the two conjectures. Putting all these implications together, we obtain the pleasant fact that the logarithmically averaged Sarnak and Chowla conjectures are equivalent, which is the main result of the current paper. There have been a large number of special cases of the Sarnak conjecture worked out (when the deterministic sequence involved came from a special dynamical system), so these results can now also be viewed as partial progress towards the Chowla conjecture also (at least with logarithmic averaging). However, my feeling is that the full resolution of these conjectures will not come from these sorts of special cases; instead, conjectures like the Fourier-uniformity conjecture in this previous post look more promising to attack.

It would also be nice to get rid of the pesky logarithmic averaging, but this seems to be an inherent requirement of the entropy decrement argument method, so one would probably have to find a way to avoid that argument if one were to remove the log averaging.

Over the last few years, a large group of mathematicians have been developing an online database to systematically collect the known facts, numerical data, and algorithms concerning some of the most central types of objects in modern number theory, namely the L-functions associated to various number fields, curves, and modular forms, as well as further data about these modular forms.  This of course includes the most famous examples of L-functions and modular forms respectively, namely the Riemann zeta function \zeta(s) and the discriminant modular form \Delta(q), but there are countless other examples of both. The connections between these classes of objects lie at the heart of the Langlands programme.

As of today, the “L-functions and modular forms database” is now out of beta, and open to the public; at present the database is mostly geared towards specialists in computational number theory, but will hopefully develop into a more broadly useful resource as time develops.  An article by John Cremona summarising the purpose of the database can be found here.

(Thanks to Andrew Sutherland and Kiran Kedlaya for the information.)

Tamar Ziegler and I have just uploaded to the arXiv two related papers: “Concatenation theorems for anti-Gowers-uniform functions and Host-Kra characteoristic factors” and “polynomial patterns in primes“, with the former developing a “quantitative Bessel inequality” for local Gowers norms that is crucial in the latter.

We use the term “concatenation theorem” to denote results in which structural control of a function in two or more “directions” can be “concatenated” into structural control in a joint direction. A trivial example of such a concatenation theorem is the following: if a function {f: {\bf Z} \times {\bf Z} \rightarrow {\bf R}} is constant in the first variable (thus {x \mapsto f(x,y)} is constant for each {y}), and also constant in the second variable (thus {y \mapsto f(x,y)} is constant for each {x}), then it is constant in the joint variable {(x,y)}. A slightly less trivial example: if a function {f: {\bf Z} \times {\bf Z} \rightarrow {\bf R}} is affine-linear in the first variable (thus, for each {y}, there exist {\alpha(y), \beta(y)} such that {f(x,y) = \alpha(y) x + \beta(y)} for all {x}) and affine-linear in the second variable (thus, for each {x}, there exist {\gamma(x), \delta(x)} such that {f(x,y) = \gamma(x)y + \delta(x)} for all {y}) then {f} is a quadratic polynomial in {x,y}; in fact it must take the form

\displaystyle f(x,y) = \epsilon xy + \zeta x + \eta y + \theta \ \ \ \ \ (1)

 

for some real numbers {\epsilon, \zeta, \eta, \theta}. (This can be seen for instance by using the affine linearity in {y} to show that the coefficients {\alpha(y), \beta(y)} are also affine linear.)

The same phenomenon extends to higher degree polynomials. Given a function {f: G \rightarrow K} from one additive group {G} to another, we say that {f} is of degree less than {d} along a subgroup {H} of {G} if all the {d}-fold iterated differences of {f} along directions in {H} vanish, that is to say

\displaystyle \partial_{h_1} \dots \partial_{h_d} f(x) = 0

for all {x \in G} and {h_1,\dots,h_d \in H}, where {\partial_h} is the difference operator

\displaystyle \partial_h f(x) := f(x+h) - f(x).

(We adopt the convention that the only {f} of degree less than {0} is the zero function.)

We then have the following simple proposition:

Proposition 1 (Concatenation of polynomiality) Let {f: G \rightarrow K} be of degree less than {d_1} along one subgroup {H_1} of {G}, and of degree less than {d_2} along another subgroup {H_2} of {G}, for some {d_1,d_2 \geq 1}. Then {f} is of degree less than {d_1+d_2-1} along the subgroup {H_1+H_2} of {G}.

Note the previous example was basically the case when {G = {\bf Z} \times {\bf Z}}, {H_1 = {\bf Z} \times \{0\}}, {H_2 = \{0\} \times {\bf Z}}, {K = {\bf R}}, and {d_1=d_2=2}.

Proof: The claim is trivial for {d_1=1} or {d_2=1} (in which {f} is constant along {H_1} or {H_2} respectively), so suppose inductively {d_1,d_2 \geq 2} and the claim has already been proven for smaller values of {d_1-1}.

We take a derivative in a direction {h_1 \in H_1} along {h_1} to obtain

\displaystyle T^{-h_1} f = f + \partial_{h_1} f

where {T^{-h_1} f(x) = f(x+h_1)} is the shift of {f} by {-h_1}. Then we take a further shift by a direction {h_2 \in H_2} to obtain

\displaystyle T^{-h_1-h_2} f = T^{-h_2} f + T^{-h_2} \partial_{h_1} f = f + \partial_{h_2} f + T^{-h_2} \partial_{h_1} f

leading to the cocycle equation

\displaystyle \partial_{h_1+h_2} f = \partial_{h_2} f + T^{-h_2} \partial_{h_1} f.

Since {f} has degree less than {d_1} along {H_1} and degree less than {d_2} along {H_2}, {\partial_{h_1} f} has degree less than {d_1-1} along {H_1} and less than {d_2} along {H_2}, so is degree less than {d_1+d_2-2} along {H_1+H_2} by induction hypothesis. Similarly {\partial_{h_2} f} is also of degree less than {d_1+d_2-2} along {H_1+H_2}. Combining this with the cocycle equation we see that {\partial_{h_1+h_2}f} is of degree less than {d_1+d_2-2} along {H_1+H_2} for any {h_1+h_2 \in H_1+H_2}, and hence {f} is of degree less than {d_1+d_2-1} along {H_1+H_2}, as required. \Box

While this proposition is simple, it already illustrates some basic principles regarding how one would go about proving a concatenation theorem:

  • (i) One should perform induction on the degrees {d_1,d_2} involved, and take advantage of the recursive nature of degree (in this case, the fact that a function is of less than degree {d} along some subgroup {H} of directions iff all of its first derivatives along {H} are of degree less than {d-1}).
  • (ii) Structure is preserved by operations such as addition, shifting, and taking derivatives. In particular, if a function {f} is of degree less than {d} along some subgroup {H}, then any derivative {\partial_k f} of {f} is also of degree less than {d} along {H}, even if {k} does not belong to {H}.

Here is another simple example of a concatenation theorem. Suppose an at most countable additive group {G} acts by measure-preserving shifts {T: g \mapsto T^g} on some probability space {(X, {\mathcal X}, \mu)}; we call the pair {(X,T)} (or more precisely {(X, {\mathcal X}, \mu, T)}) a {G}-system. We say that a function {f \in L^\infty(X)} is a generalised eigenfunction of degree less than {d} along some subgroup {H} of {G} and some {d \geq 1} if one has

\displaystyle T^h f = \lambda_h f

almost everywhere for all {h \in H}, and some functions {\lambda_h \in L^\infty(X)} of degree less than {d-1} along {H}, with the convention that a function has degree less than {0} if and only if it is equal to {1}. Thus for instance, a function {f} is an generalised eigenfunction of degree less than {1} along {H} if it is constant on almost every {H}-ergodic component of {G}, and is a generalised function of degree less than {2} along {H} if it is an eigenfunction of the shift action on almost every {H}-ergodic component of {G}. A basic example of a higher order eigenfunction is the function {f(x,y) := e^{2\pi i y}} on the skew shift {({\bf R}/{\bf Z})^2} with {{\bf Z}} action given by the generator {T(x,y) := (x+\alpha,y+x)} for some irrational {\alpha}. One can check that {T^h f = \lambda_h f} for every integer {h}, where {\lambda_h: x \mapsto e^{2\pi i \binom{h}{2} \alpha} e^{2\pi i h x}} is a generalised eigenfunction of degree less than {2} along {{\bf Z}}, so {f} is of degree less than {3} along {{\bf Z}}.

We then have

Proposition 2 (Concatenation of higher order eigenfunctions) Let {(X,T)} be a {G}-system, and let {f \in L^\infty(X)} be a generalised eigenfunction of degree less than {d_1} along one subgroup {H_1} of {G}, and a generalised eigenfunction of degree less than {d_2} along another subgroup {H_2} of {G}, for some {d_1,d_2 \geq 1}. Then {f} is a generalised eigenfunction of degree less than {d_1+d_2-1} along the subgroup {H_1+H_2} of {G}.

The argument is almost identical to that of the previous proposition and is left as an exercise to the reader. The key point is the point (ii) identified earlier: the space of generalised eigenfunctions of degree less than {d} along {H} is preserved by multiplication and shifts, as well as the operation of “taking derivatives” {f \mapsto \lambda_k} even along directions {k} that do not lie in {H}. (To prove this latter claim, one should restrict to the region where {f} is non-zero, and then divide {T^k f} by {f} to locate {\lambda_k}.)

A typical example of this proposition in action is as follows: consider the {{\bf Z}^2}-system given by the {3}-torus {({\bf R}/{\bf Z})^3} with generating shifts

\displaystyle T^{(1,0)}(x,y,z) := (x+\alpha,y,z+y)

\displaystyle T^{(0,1)}(x,y,z) := (x,y+\alpha,z+x)

for some irrational {\alpha}, which can be checked to give a {{\bf Z}^2} action

\displaystyle T^{(n,m)}(x,y,z) := (x+n\alpha, y+m\alpha, z+ny+mx+nm\alpha).

The function {f(x,y,z) := e^{2\pi i z}} can then be checked to be a generalised eigenfunction of degree less than {2} along {{\bf Z} \times \{0\}}, and also less than {2} along {\{0\} \times {\bf Z}}, and less than {3} along {{\bf Z}^2}. One can view this example as the dynamical systems translation of the example (1) (see this previous post for some more discussion of this sort of correspondence).

The main results of our concatenation paper are analogues of these propositions concerning a more complicated notion of “polynomial-like” structure that are of importance in additive combinatorics and in ergodic theory. On the ergodic theory side, the notion of structure is captured by the Host-Kra characteristic factors {Z^{<d}_H(X)} of a {G}-system {X} along a subgroup {H}. These factors can be defined in a number of ways. One is by duality, using the Gowers-Host-Kra uniformity seminorms (defined for instance here) {\| \|_{U^d_H(X)}}. Namely, {Z^{<d}_H(X)} is the factor of {X} defined up to equivalence by the requirement that

\displaystyle \|f\|_{U^d_H(X)} = 0 \iff {\bf E}(f | Z^{<d}_H(X) ) = 0.

An equivalent definition is in terms of the dual functions {{\mathcal D}^d_H(f)} of {f} along {H}, which can be defined recursively by setting {{\mathcal D}^0_H(f) = 1} and

\displaystyle {\mathcal D}^d_H(f) = {\bf E}_h T^h f {\mathcal D}^{d-1}( f \overline{T^h f} )

where {{\bf E}_h} denotes the ergodic average along a Følner sequence in {G} (in fact one can also define these concepts in non-amenable abelian settings as per this previous post). The factor {Z^{<d}_H(X)} can then be alternately defined as the factor generated by the dual functions {{\mathcal D}^d_H(f)} for {f \in L^\infty(X)}.

In the case when {G=H={\bf Z}} and {X} is {G}-ergodic, a deep theorem of Host and Kra shows that the factor {Z^{<d}_H(X)} is equivalent to the inverse limit of nilsystems of step less than {d}. A similar statement holds with {{\bf Z}} replaced by any finitely generated group by Griesmer, while the case of an infinite vector space over a finite field was treated in this paper of Bergelson, Ziegler, and myself. The situation is more subtle when {X} is not {G}-ergodic, or when {X} is {G}-ergodic but {H} is a proper subgroup of {G} acting non-ergodically, when one has to start considering measurable families of directional nilsystems; see for instance this paper of Austin for some of the subtleties involved (for instance, higher order group cohomology begins to become relevant!).

One of our main theorems is then

Proposition 3 (Concatenation of characteristic factors) Let {(X,T)} be a {G}-system, and let {f} be measurable with respect to the factor {Z^{<d_1}_{H_1}(X)} and with respect to the factor {Z^{<d_2}_{H_2}(X)} for some {d_1,d_2 \geq 1} and some subgroups {H_1,H_2} of {G}. Then {f} is also measurable with respect to the factor {Z^{<d_1+d_2-1}_{H_1+H_2}(X)}.

We give two proofs of this proposition in the paper; an ergodic-theoretic proof using the Host-Kra theory of “cocycles of type {<d} (along a subgroup {H})”, which can be used to inductively describe the factors {Z^{<d}_H}, and a combinatorial proof based on a combinatorial analogue of this proposition which is harder to state (but which roughly speaking asserts that a function which is nearly orthogonal to all bounded functions of small {U^{d_1}_{H_1}} norm, and also to all bounded functions of small {U^{d_2}_{H_2}} norm, is also nearly orthogonal to alll bounded functions of small {U^{d_1+d_2-1}_{H_1+H_2}} norm). The combinatorial proof parallels the proof of Proposition 2. A key point is that dual functions {F := {\mathcal D}^d_H(f)} obey a property analogous to being a generalised eigenfunction, namely that

\displaystyle T^h F = {\bf E}_k \lambda_{h,k} F_k

where {F_k := T^k F} and {\lambda_{h,k} := {\mathcal D}^{d-1}( T^h f \overline{T^k f} )} is a “structured function of order {d-1}” along {H}. (In the language of this previous paper of mine, this is an assertion that dual functions are uniformly almost periodic of order {d}.) Again, the point (ii) above is crucial, and in particular it is key that any structure that {F} has is inherited by the associated functions {\lambda_{h,k}} and {F_k}. This sort of inheritance is quite easy to accomplish in the ergodic setting, as there is a ready-made language of factors to encapsulate the concept of structure, and the shift-invariance and {\sigma}-algebra properties of factors make it easy to show that just about any “natural” operation one performs on a function measurable with respect to a given factor, returns a function that is still measurable in that factor. In the finitary combinatorial setting, though, encoding the fact (ii) becomes a remarkably complicated notational nightmare, requiring a huge amount of “epsilon management” and “second-order epsilon management” (in which one manages not only scalar epsilons, but also function-valued epsilons that depend on other parameters). In order to avoid all this we were forced to utilise a nonstandard analysis framework for the combinatorial theorems, which made the arguments greatly resemble the ergodic arguments in many respects (though the two settings are still not equivalent, see this previous blog post for some comparisons between the two settings). Unfortunately the arguments are still rather complicated.

For combinatorial applications, dual formulations of the concatenation theorem are more useful. A direct dualisation of the theorem yields the following decomposition theorem: a bounded function which is small in {U^{d_1+d_2-1}_{H_1+H_2}} norm can be split into a component that is small in {U^{d_1}_{H_1}} norm, and a component that is small in {U^{d_2}_{H_2}} norm. (One may wish to understand this type of result by first proving the following baby version: any function that has mean zero on every coset of {H_1+H_2}, can be decomposed as the sum of a function that has mean zero on every {H_1} coset, and a function that has mean zero on every {H_2} coset. This is dual to the assertion that a function that is constant on every {H_1} coset and constant on every {H_2} coset, is constant on every {H_1+H_2} coset.) Combining this with some standard “almost orthogonality” arguments (i.e. Cauchy-Schwarz) give the following Bessel-type inequality: if one has a lot of subgroups {H_1,\dots,H_k} and a bounded function is small in {U^{2d-1}_{H_i+H_j}} norm for most {i,j}, then it is also small in {U^d_{H_i}} norm for most {i}. (Here is a baby version one may wish to warm up on: if a function {f} has small mean on {({\bf Z}/p{\bf Z})^2} for some large prime {p}, then it has small mean on most of the cosets of most of the one-dimensional subgroups of {({\bf Z}/p{\bf Z})^2}.)

There is also a generalisation of the above Bessel inequality (as well as several of the other results mentioned above) in which the subgroups {H_i} are replaced by more general coset progressions {H_i+P_i} (of bounded rank), so that one has a Bessel inequailty controlling “local” Gowers uniformity norms such as {U^d_{P_i}} by “global” Gowers uniformity norms such as {U^{2d-1}_{P_i+P_j}}. This turns out to be particularly useful when attempting to compute polynomial averages such as

\displaystyle \sum_{n \leq N} \sum_{r \leq \sqrt{N}} f(n) g(n+r^2) h(n+2r^2) \ \ \ \ \ (2)

 

for various functions {f,g,h}. After repeated use of the van der Corput lemma, one can control such averages by expressions such as

\displaystyle \sum_{n \leq N} \sum_{h,m,k \leq \sqrt{N}} f(n) f(n+mh) f(n+mk) f(n+m(h+k))

(actually one ends up with more complicated expressions than this, but let’s use this example for sake of discussion). This can be viewed as an average of various {U^2} Gowers uniformity norms of {f} along arithmetic progressions of the form {\{ mh: h \leq \sqrt{N}\}} for various {m \leq \sqrt{N}}. Using the above Bessel inequality, this can be controlled in turn by an average of various {U^3} Gowers uniformity norms along rank two generalised arithmetic progressions of the form {\{ m_1 h_1 + m_2 h_2: h_1,h_2 \le \sqrt{N}\}} for various {m_1,m_2 \leq \sqrt{N}}. But for generic {m_1,m_2}, this rank two progression is close in a certain technical sense to the “global” interval {\{ n: n \leq N \}} (this is ultimately due to the basic fact that two randomly chosen large integers are likely to be coprime, or at least have a small gcd). As a consequence, one can use the concatenation theorems from our first paper to control expressions such as (2) in terms of global Gowers uniformity norms. This is important in number theoretic applications, when one is interested in computing sums such as

\displaystyle \sum_{n \leq N} \sum_{r \leq \sqrt{N}} \mu(n) \mu(n+r^2) \mu(n+2r^2)

or

\displaystyle \sum_{n \leq N} \sum_{r \leq \sqrt{N}} \Lambda(n) \Lambda(n+r^2) \Lambda(n+2r^2)

where {\mu} and {\Lambda} are the Möbius and von Mangoldt functions respectively. This is because we are able to control global Gowers uniformity norms of such functions (thanks to results such as the proof of the inverse conjecture for the Gowers norms, the orthogonality of the Möbius function with nilsequences, and asymptotics for linear equations in primes), but much less control is currently available for local Gowers uniformity norms, even with the assistance of the generalised Riemann hypothesis (see this previous blog post for some further discussion).

By combining these tools and strategies with the “transference principle” approach from our previous paper (as improved using the recent “densification” technique of Conlon, Fox, and Zhao, discussed in this previous post), we are able in particular to establish the following result:

Theorem 4 (Polynomial patterns in the primes) Let {P_1,\dots,P_k: {\bf Z} \rightarrow {\bf Z}} be polynomials of degree at most {d}, whose degree {d} coefficients are all distinct, for some {d \geq 1}. Suppose that {P_1,\dots,P_k} is admissible in the sense that for every prime {p}, there are {n,r} such that {n+P_1(r),\dots,n+P_k(r)} are all coprime to {p}. Then there exist infinitely many pairs {n,r} of natural numbers such that {n+P_1(r),\dots,n+P_k(r)} are prime.

Furthermore, we obtain an asymptotic for the number of such pairs {n,r} in the range {n \leq N}, {r \leq N^{1/d}} (actually for minor technical reasons we reduce the range of {r} to be very slightly less than {N^{1/d}}). In fact one could in principle obtain asymptotics for smaller values of {r}, and relax the requirement that the degree {d} coefficients be distinct with the requirement that no two of the {P_i} differ by a constant, provided one had good enough local uniformity results for the Möbius or von Mangoldt functions. For instance, we can obtain an asymptotic for triplets of the form {n, n+r,n+r^d} unconditionally for {d \leq 5}, and conditionally on GRH for all {d}, using known results on primes in short intervals on average.

The {d=1} case of this theorem was obtained in a previous paper of myself and Ben Green (using the aforementioned conjectures on the Gowers uniformity norm and the orthogonality of the Möbius function with nilsequences, both of which are now proven). For higher {d}, an older result of Tamar and myself was able to tackle the case when {P_1(0)=\dots=P_k(0)=0} (though our results there only give lower bounds on the number of pairs {(n,r)}, and no asymptotics). Both of these results generalise my older theorem with Ben Green on the primes containing arbitrarily long arithmetic progressions. The theorem also extends to multidimensional polynomials, in which case there are some additional previous results; see the paper for more details. We also get a technical refinement of our previous result on narrow polynomial progressions in (dense subsets of) the primes by making the progressions just a little bit narrower in the case of the density of the set one is using is small.

. This latter Bessel type inequality is particularly useful in combinatorial and number-theoretic applications, as it allows one to convert “global” Gowers uniformity norm (basically, bounds on norms such as {U^{2d-1}_{H_i+H_j}}) to “local” Gowers uniformity norm control.

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