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Kaisa Matomäki, Xuancheng Shao, Joni Teräväinen, and myself have just uploaded to the arXiv our preprint “Higher uniformity of arithmetic functions in short intervals I. All intervals“. This paper investigates the higher order (Gowers) uniformity of standard arithmetic functions in analytic number theory (and specifically, the Möbius function , the von Mangoldt function
, and the generalised divisor functions
) in short intervals
, where
is large and
lies in the range
for a fixed constant
(that one would like to be as small as possible). If we let
denote one of the functions
, then there is extensive literature on the estimation of short sums
Traditionally, asymptotics for such sums are expressed in terms of a “main term” of some arithmetic nature, plus an error term that is estimated in magnitude. For instance, a sum such as would be approximated in terms of a main term that vanished (or is negligible) if
is “minor arc”, but would be expressible in terms of something like a Ramanujan sum if
was “major arc”, together with an error term. We found it convenient to cancel off such main terms by subtracting an approximant
from each of the arithmetic functions
and then getting upper bounds on remainder correlations such as
- For the Möbius function
, we simply set
, as per the Möbius pseudorandomness conjecture. (One could choose a more sophisticated approximant in the presence of a Siegel zero, as I did with Joni in this recent paper, but we do not do so here.)
- For the von Mangoldt function
, we eventually went with the Cramér-Granville approximant
, where
and
.
- For the divisor functions
, we used a somewhat complicated-looking approximant
for some explicit polynomials
, chosen so that
and
have almost exactly the same sums along arithmetic progressions (see the paper for details).
The objective is then to obtain bounds on sums such as (1) that improve upon the “trivial bound” that one can get with the triangle inequality and standard number theory bounds such as the Brun-Titchmarsh inequality. For and
, the Siegel-Walfisz theorem suggests that it is reasonable to expect error terms that have “strongly logarithmic savings” in the sense that they gain a factor of
over the trivial bound for any
; for
, the Dirichlet hyperbola method suggests instead that one has “power savings” in that one should gain a factor of
over the trivial bound for some
. In the case of the Möbius function
, there is an additional trick (introduced by Matomäki and Teräväinen) that allows one to lower the exponent
somewhat at the cost of only obtaining “weakly logarithmic savings” of shape
for some small
.
Our main estimates on sums of the form (1) work in the following ranges:
- For
, one can obtain strongly logarithmic savings on (1) for
, and power savings for
.
- For
, one can obtain weakly logarithmic savings for
.
- For
, one can obtain power savings for
.
- For
, one can obtain power savings for
.
Conjecturally, one should be able to obtain power savings in all cases, and lower down to zero, but the ranges of exponents and savings given here seem to be the limit of current methods unless one assumes additional hypotheses, such as GRH. The
result for correlation against Fourier phases
was established previously by Zhan, and the
result for such phases and
was established previously by by Matomäki and Teräväinen.
By combining these results with tools from additive combinatorics, one can obtain a number of applications:
- Direct insertion of our bounds in the recent work of Kanigowski, Lemanczyk, and Radziwill on the prime number theorem on dynamical systems that are analytic skew products gives some improvements in the exponents there.
- We can obtain a “short interval” version of a multiple ergodic theorem along primes established by Frantzikinakis-Host-Kra and Wooley-Ziegler, in which we average over intervals of the form
rather than
.
- We can obtain a “short interval” version of the “linear equations in primes” asymptotics obtained by Ben Green, Tamar Ziegler, and myself in this sequence of papers, where the variables in these equations lie in short intervals
rather than long intervals such as
.
We now briefly discuss some of the ingredients of proof of our main results. The first step is standard, using combinatorial decompositions (based on the Heath-Brown identity and (for the result) the Ramaré identity) to decompose
into more tractable sums of the following types:
- Type
sums, which are basically of the form
for some weights
of controlled size and some cutoff
that is not too large;
- Type
sums, which are basically of the form
for some weights
,
of controlled size and some cutoffs
that are not too close to
or to
;
- Type
sums, which are basically of the form
for some weights
of controlled size and some cutoff
that is not too large.
The precise ranges of the cutoffs depend on the choice of
; our methods fail once these cutoffs pass a certain threshold, and this is the reason for the exponents
being what they are in our main results.
The Type sums involving nilsequences can be treated by methods similar to those in this previous paper of Ben Green and myself; the main innovations are in the treatment of the Type
and Type
sums.
For the Type sums, one can split into the “abelian” case in which (after some Fourier decomposition) the nilsequence
is basically of the form
, and the “non-abelian” case in which
is non-abelian and
exhibits non-trivial oscillation in a central direction. In the abelian case we can adapt arguments of Matomaki and Shao, which uses Cauchy-Schwarz and the equidistribution properties of polynomials to obtain good bounds unless
is “major arc” in the sense that it resembles (or “pretends to be”)
for some Dirichlet character
and some frequency
, but in this case one can use classical multiplicative methods to control the correlation. It turns out that the non-abelian case can be treated similarly. After applying Cauchy-Schwarz, one ends up analyzing the equidistribution of the four-variable polynomial sequence
For the type sum, a model sum to study is
In a sequel to this paper (currently in preparation), we will obtain analogous results for almost all intervals with
in the range
, in which we will be able to lower
all the way to
.
Joni Teräväinen and I have just uploaded to the arXiv our preprint “The Hardy–Littlewood–Chowla conjecture in the presence of a Siegel zero“. This paper is a development of the theme that certain conjectures in analytic number theory become easier if one makes the hypothesis that Siegel zeroes exist; this places one in a presumably “illusory” universe, since the widely believed Generalised Riemann Hypothesis (GRH) precludes the existence of such zeroes, yet this illusory universe seems remarkably self-consistent and notoriously impossible to eliminate from one’s analysis.
For the purposes of this paper, a Siegel zero is a zero of a Dirichlet
-function
corresponding to a primitive quadratic character
of some conductor
, which is close to
in the sense that
One of the early influential results in this area was the following result of Heath-Brown, which I previously blogged about here:
Theorem 1 (Hardy-Littlewood assuming Siegel zero) Letbe a fixed natural number. Suppose one has a Siegel zero
associated to some conductor
. Then we have
for all
, where
is the von Mangoldt function and
is the singular series
In particular, Heath-Brown showed that if there are infinitely many Siegel zeroes, then there are also infinitely many twin primes, with the correct asymptotic predicted by the Hardy-Littlewood prime tuple conjecture at infinitely many scales.
Very recently, Chinis established an analogous result for the Chowla conjecture (building upon earlier work of Germán and Katai):
Theorem 2 (Chowla assuming Siegel zero) Letbe distinct fixed natural numbers. Suppose one has a Siegel zero
associated to some conductor
. Then one has
in the range
, where
is the Liouville function.
In our paper we unify these results and also improve the quantitative estimates and range of :
Theorem 3 (Hardy-Littlewood-Chowla assuming Siegel zero) Letbe distinct fixed natural numbers with
. Suppose one has a Siegel zero
associated to some conductor
. Then one has
for
for any fixed
.
Our argument proceeds by a series of steps in which we replace and
by more complicated looking, but also more tractable, approximations, until the correlation is one that can be computed in a tedious but straightforward fashion by known techniques. More precisely, the steps are as follows:
- (i) Replace the Liouville function
with an approximant
, which is a completely multiplicative function that agrees with
at small primes and agrees with
at large primes.
- (ii) Replace the von Mangoldt function
with an approximant
, which is the Dirichlet convolution
multiplied by a Selberg sieve weight
to essentially restrict that convolution to almost primes.
- (iii) Replace
with a more complicated truncation
which has the structure of a “Type I sum”, and which agrees with
on numbers that have a “typical” factorization.
- (iv) Replace the approximant
with a more complicated approximant
which has the structure of a “Type I sum”.
- (v) Now that all terms in the correlation have been replaced with tractable Type I sums, use standard Euler product calculations and Fourier analysis, similar in spirit to the proof of the pseudorandomness of the Selberg sieve majorant for the primes in this paper of Ben Green and myself, to evaluate the correlation to high accuracy.
Steps (i), (ii) proceed mainly through estimates such as (1) and standard sieve theory bounds. Step (iii) is based primarily on estimates on the number of smooth numbers of a certain size.
The restriction in our main theorem is needed only to execute step (iv) of this step. Roughly speaking, the Siegel approximant
to
is a twisted, sieved version of the divisor function
, and the types of correlation one is faced with at the start of step (iv) are a more complicated version of the divisor correlation sum
Step (v) is a tedious but straightforward sieve theoretic computation, similar in many ways to the correlation estimates of Goldston and Yildirim used in their work on small gaps between primes (as discussed for instance here), and then also used by Ben Green and myself to locate arithmetic progressions in primes.
Joni Teräväinen and myself have just uploaded to the arXiv our preprint “Quantitative bounds for Gowers uniformity of the Möbius and von Mangoldt functions“. This paper makes quantitative the Gowers uniformity estimates on the Möbius function and the von Mangoldt function
.
To discuss the results we first discuss the situation of the Möbius function, which is technically simpler in some (though not all) ways. We assume familiarity with Gowers norms and standard notations around these norms, such as the averaging notation and the exponential notation
. The prime number theorem in qualitative form asserts that
Once one restricts to arithmetic progressions, the situation gets worse: the Siegel-Walfisz theorem gives the bound
for any residue classIn 1937, Davenport was able to show the discorrelation estimate
For the situation with the norm the previously known results were much weaker. Ben Green and I showed that
For higher norms , the situation is even worse, because the quantitative inverse theory for these norms is poorer, and indeed it was only with the recent work of Manners that any such bound is available at all (at least for
). Basically, Manners establishes if
Our first result gives an effective decay bound:
Theorem 1 For any, we have
for some
. The implied constants are effective.
This is off by a logarithm from the best effective bound (2) in the case. In the
case there is some hope to remove this logarithm based on the improved quantitative inverse theory currently available in this case, but there is a technical obstruction to doing so which we will discuss later in this post. For
the above bound is the best one could hope to achieve purely using the quantitative inverse theory of Manners.
We have analogues of all the above results for the von Mangoldt function . Here a complication arises that
does not have mean close to zero, and one has to subtract off some suitable approximant
to
before one would expect good Gowers norms bounds. For the prime number theorem one can just use the approximant
, giving
Theorem 2 For any, we have
for some
. The implied constants are effective.
By standard methods, this result also gives quantitative asymptotics for counting solutions to various systems of linear equations in primes, with error terms that gain a factor of with respect to the main term.
We now discuss the methods of proof, focusing first on the case of the Möbius function. Suppose first that there is no “Siegel zero”, by which we mean a quadratic character of some conductor
with a zero
with
for some small absolute constant
. In this case the Siegel-Walfisz bound (1) improves to a quasipolynomial bound
Now suppose we have a Siegel zero . In this case the bound (5) will not hold in general, and hence also (6) will not hold either. Here, the usual way out (while still maintaining effective estimates) is to approximate
not by
, but rather by a more complicated approximant
that takes the Siegel zero into account, and in particular is such that one has the (effective) pseudopolynomial bound
For the analogous problem with the von Mangoldt function (assuming a Siegel zero for sake of discussion), the approximant is simpler; we ended up using
In principle, the above results can be improved for due to the stronger quantitative inverse theorems in the
setting. However, there is a bottleneck that prevents us from achieving this, namely that the equidistribution theory of two-step nilmanifolds has exponents which are exponential in the dimension rather than polynomial in the dimension, and as a consequence we were unable to improve upon the doubly logarithmic results. Specifically, if one is given a sequence of bracket quadratics such as
that fails to be
-equidistributed, one would need to establish a nontrivial linear relationship modulo 1 between the
(up to errors of
), where the coefficients are of size
; current methods only give coefficient bounds of the form
. An old result of Schmidt demonstrates proof of concept that these sorts of polynomial dependencies on exponents is possible in principle, but actually implementing Schmidt’s methods here seems to be a quite non-trivial task. There is also another possible route to removing a logarithm, which is to strengthen the inverse
theorem to make the dimension of the nilmanifold logarithmic in the uniformity parameter
rather than polynomial. Again, the Freiman-Bilu theorem (see for instance this paper of Ben and myself) demonstrates proof of concept that such an improvement in dimension is possible, but some work would be needed to implement it.
Kaisa Matomäki, Maksym Radziwill, Xuancheng Shao, Joni Teräväinen, and myself have just uploaded to the arXiv our preprint “Singmaster’s conjecture in the interior of Pascal’s triangle“. This paper leverages the theory of exponential sums over primes to make progress on a well known conjecture of Singmaster which asserts that any natural number larger than appears at most a bounded number of times in Pascal’s triangle. That is to say, for any integer
, there are at most
solutions to the equation
Our main result settles this conjecture in the “interior” region of the triangle:
Theorem 1 (Singmaster’s conjecture in the interior of the triangle) Ifand
is sufficiently large depending on
, there are at most two solutions to (1) in the region
and hence at most four in the region
Also, there is at most one solution in the region
To verify Singmaster’s conjecture in full, it thus suffices in view of this result to verify the conjecture in the boundary region
(or equivalentlyThe upper bound of two here for the number of solutions in the region (2) is best possible, due to the infinite family of solutions to the equation
coming from
The appearance of the quantity in Theorem 1 may be familiar to readers that are acquainted with Vinogradov’s bounds on exponential sums, which ends up being the main new ingredient in our arguments. In principle this threshold could be lowered if we had stronger bounds on exponential sums.
To try to control solutions to (1) we use a combination of “Archimedean” and “non-Archimedean” approaches. In the “Archimedean” approach (following earlier work of Kane on this problem) we view primarily as real numbers rather than integers, and express (1) in terms of the Gamma function as
Proposition 2 Let, and suppose
is sufficiently large depending on
. If
is a solution to (1) in the left half
of Pascal’s triangle, then there is at most one other solution
to this equation in the left half with
Again, the example of (4) shows that a cluster of two solutions is certainly possible; the convexity argument only kicks in once one has a cluster of three or more solutions.
To finish the proof of Theorem 1, one has to show that any two solutions to (1) in the region of interest must be close enough for the above proposition to apply. Here we switch to the “non-Archimedean” approach, in which we look at the
-adic valuations
of the binomial coefficients, defined as the number of times a prime
divides
. From the fundamental theorem of arithmetic, a collision
A key idea in our approach is to view this condition (6) statistically, for instance by viewing as a prime drawn randomly from an interval such as
for some suitably chosen scale parameter
, so that the two sides of (6) now become random variables. It then becomes advantageous to compare correlations between these two random variables and some additional test random variable. For instance, if
and
are far apart from each other, then one would expect the left-hand side of (6) to have a higher correlation with the fractional part
, since this term shows up in the summation on the left-hand side but not the right. Similarly if
and
are far apart from each other (although there are some annoying cases one has to treat separately when there is some “unexpected commensurability”, for instance if
is a rational multiple of
where the rational has bounded numerator and denominator). In order to execute this strategy, it turns out (after some standard Fourier expansion) that one needs to get good control on exponential sums such as
A modification of the arguments also gives similar results for the equation
where
Theorem 3 Ifand
is sufficiently large depending on
, there are at most two solutions to (7) in the region
Again the upper bound of two is best possible, thanks to identities such as
Kaisa Matomäki, Maksym Radziwill, Joni Teräväinen, Tamar Ziegler and I have uploaded to the arXiv our paper Higher uniformity of bounded multiplicative functions in short intervals on average. This paper (which originated from a working group at an AIM workshop on Sarnak’s conjecture) focuses on the local Fourier uniformity conjecture for bounded multiplicative functions such as the Liouville function . One form of this conjecture is the assertion that
The conjecture gets more difficult as increases, and also becomes more difficult the more slowly
grows with
. The
conjecture is equivalent to the assertion
For , the conjecture is equivalent to the assertion
Now we apply the same strategy to (4). For abelian the claim follows easily from (3), so we focus on the non-abelian case. One now has a polynomial sequence
attached to many
, and after a somewhat complicated adaptation of the above arguments one again ends up with an approximate functional equation
We give two applications of this higher order Fourier uniformity. One regards the growth of the number
The second application is to obtain cancellation for various polynomial averages involving the Liouville function or von Mangoldt function
, such as
Joni Teräväinen and I have just uploaded to the arXiv our paper “Value patterns of multiplicative functions and related sequences“, submitted to Forum of Mathematics, Sigma. This paper explores how to use recent technology on correlations of multiplicative (or nearly multiplicative functions), such as the “entropy decrement method”, in conjunction with techniques from additive combinatorics, to establish new results on the sign patterns of functions such as the Liouville function . For instance, with regards to length 5 sign patterns
of the Liouville function, we can now show that at least of the
possible sign patterns in
occur with positive upper density. (Conjecturally, all of them do so, and this is known for all shorter sign patterns, but unfortunately
seems to be the limitation of our methods.)
The Liouville function can be written as , where
is the number of prime factors of
(counting multiplicity). One can also consider the variant
, which is a completely multiplicative function taking values in the cube roots of unity
. Here we are able to show that all
sign patterns in
occur with positive lower density as sign patterns
of this function. The analogous result for
was already known (see this paper of Matomäki, Radziwiłł, and myself), and in that case it is even known that all sign patterns occur with equal logarithmic density
(from this paper of myself and Teräväinen), but these techniques barely fail to handle the
case by itself (largely because the “parity” arguments used in the case of the Liouville function no longer control three-point correlations in the
case) and an additional additive combinatorial tool is needed. After applying existing technology (such as entropy decrement methods), the problem roughly speaking reduces to locating patterns
for a certain partition
of a compact abelian group
(think for instance of the unit circle
, although the general case is a bit more complicated, in particular if
is disconnected then there is a certain “coprimality” constraint on
, also we can allow the
to be replaced by any
with
divisible by
), with each of the
having measure
. An inequality of Kneser just barely fails to guarantee the existence of such patterns, but by using an inverse theorem for Kneser’s inequality in this previous paper of mine we are able to identify precisely the obstruction for this method to work, and rule it out by an ad hoc method.
The same techniques turn out to also make progress on some conjectures of Erdös-Pomerance and Hildebrand regarding patterns of the largest prime factor of a natural number
. For instance, we improve results of Erdös-Pomerance and of Balog demonstrating that the inequalities
and
each hold for infinitely many , by demonstrating the stronger claims that the inequalities
and
each hold for a set of of positive lower density. As a variant, we also show that we can find a positive density set of
for which
for any fixed (this improves on a previous result of Hildebrand with
replaced by
. A number of other results of this type are also obtained in this paper.
In order to obtain these sorts of results, one needs to extend the entropy decrement technology from the setting of multiplicative functions to that of what we call “weakly stable sets” – sets which have some multiplicative structure, in the sense that (roughly speaking) there is a set
such that for all small primes
, the statements
and
are roughly equivalent to each other. For instance, if
is a level set
, one would take
; if instead
is a set of the form
, then one can take
. When one has such a situation, then very roughly speaking, the entropy decrement argument then allows one to estimate a one-parameter correlation such as
with a two-parameter correlation such as
(where we will be deliberately vague as to how we are averaging over and
), and then the use of the “linear equations in primes” technology of Ben Green, Tamar Ziegler, and myself then allows one to replace this average in turn by something like
where is constrained to be not divisible by small primes but is otherwise quite arbitrary. This latter average can then be attacked by tools from additive combinatorics, such as translation to a continuous group model (using for instance the Furstenberg correspondence principle) followed by tools such as Kneser’s inequality (or inverse theorems to that inequality).
Joni Teräväinen and I have just uploaded to the arXiv our paper “The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures“. This is a sequel to our previous paper that studied logarithmic correlations of the form
where were bounded multiplicative functions,
were fixed shifts,
was a quantity going off to infinity, and
was a generalised limit functional. Our main technical result asserted that these correlations were necessarily the uniform limit of periodic functions
. Furthermore, if
(weakly) pretended to be a Dirichlet character
, then the
could be chosen to be
–isotypic in the sense that
whenever
are integers with
coprime to the periods of
and
; otherwise, if
did not weakly pretend to be any Dirichlet character
, then
vanished completely. This was then used to verify several cases of the logarithmically averaged Elliott and Chowla conjectures.
The purpose of this paper was to investigate the extent to which the methods could be extended to non-logarithmically averaged settings. For our main technical result, we now considered the unweighted averages
where is an additional parameter. Our main result was now as follows. If
did not weakly pretend to be a twisted Dirichlet character
, then
converged to zero on (doubly logarithmic) average as
. If instead
did pretend to be such a twisted Dirichlet character, then
converged on (doubly logarithmic) average to a limit
of
-isotypic functions
. Thus, roughly speaking, one has the approximation
for most .
Informally, this says that at almost all scales (where “almost all” means “outside of a set of logarithmic density zero”), the non-logarithmic averages behave much like their logarithmic counterparts except for a possible additional twisting by an Archimedean character
(which interacts with the Archimedean parameter
in much the same way that the Dirichlet character
interacts with the non-Archimedean parameter
). One consequence of this is that most of the recent results on the logarithmically averaged Chowla and Elliott conjectures can now be extended to their non-logarithmically averaged counterparts, so long as one excludes a set of exceptional scales
of logarithmic density zero. For instance, the Chowla conjecture
is now established for either odd or equal to
, so long as one excludes an exceptional set of scales.
In the logarithmically averaged setup, the main idea was to combine two very different pieces of information on . The first, coming from recent results in ergodic theory, was to show that
was well approximated in some sense by a nilsequence. The second was to use the “entropy decrement argument” to obtain an approximate isotopy property of the form
for “most” primes and integers
. Combining the two facts, one eventually finds that only the almost periodic components of the nilsequence are relevant.
In the current situation, each is approximated by a nilsequence, but the nilsequence can vary with
(although there is some useful “Lipschitz continuity” of this nilsequence with respect to the
parameter). Meanwhile, the entropy decrement argument gives an approximation basically of the form
for “most” . The arguments then proceed largely as in the logarithmically averaged case. A key lemma to handle the dependence on the new parameter
is the following cohomological statement: if one has a map
that was a quasimorphism in the sense that
for all
and some small
, then there exists a real number
such that
for all small
. This is achieved by applying a standard “cocycle averaging argument” to the cocycle
.
It would of course be desirable to not have the set of exceptional scales. We only know of one (implausible) scenario in which we can do this, namely when one has far fewer (in particular, subexponentially many) sign patterns for (say) the Liouville function than predicted by the Chowla conjecture. In this scenario (roughly analogous to the “Siegel zero” scenario in multiplicative number theory), the entropy of the Liouville sign patterns is so small that the entropy decrement argument becomes powerful enough to control all scales rather than almost all scales. On the other hand, this scenario seems to be self-defeating, in that it allows one to establish a large number of cases of the Chowla conjecture, and the full Chowla conjecture is inconsistent with having unusually few sign patterns. Still it hints that future work in this direction may need to split into “low entropy” and “high entropy” cases, in analogy to how many arguments in multiplicative number theory have to split into the “Siegel zero” and “no Siegel zero” cases.
Joni Teräväinen and I have just uploaded to the arXiv our paper “Odd order cases of the logarithmically averaged Chowla conjecture“, submitted to J. Numb. Thy. Bordeaux. This paper gives an alternate route to one of the main results of our previous paper, and more specifically reproves the asymptotic
for all odd and all integers
(that is to say, all the odd order cases of the logarithmically averaged Chowla conjecture). Our previous argument relies heavily on some deep ergodic theory results of Bergelson-Host-Kra, Leibman, and Le (and was applicable to more general multiplicative functions than the Liouville function
); here we give a shorter proof that avoids ergodic theory (but instead requires the Gowers uniformity of the (W-tricked) von Mangoldt function, established in several papers of Ben Green, Tamar Ziegler, and myself). The proof follows the lines sketched in the previous blog post. In principle, due to the avoidance of ergodic theory, the arguments here have a greater chance to be made quantitative; however, at present the known bounds on the Gowers uniformity of the von Mangoldt function are qualitative, except at the
level, which is unfortunate since the first non-trivial odd case
requires quantitative control on the
level. (But it may be possible to make the Gowers uniformity bounds for
quantitative if one assumes GRH, although when one puts everything together, the actual decay rate obtained in (1) is likely to be poor.)
Joni Teräväinen and I have just uploaded to the arXiv our paper “The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures“, submitted to Duke Mathematical Journal. This paper builds upon my previous paper in which I introduced an “entropy decrement method” to prove the two-point (logarithmically averaged) cases of the Chowla and Elliott conjectures. A bit more specifically, I showed that
whenever were sequences going to infinity,
were distinct integers, and
were
-bounded multiplicative functions which were non-pretentious in the sense that
for all Dirichlet characters and for
. Thus, for instance, one had the logarithmically averaged two-point Chowla conjecture
for fixed any non-zero , where
was the Liouville function.
One would certainly like to extend these results to higher order correlations than the two-point correlations. This looks to be difficult (though perhaps not completely impossible if one allows for logarithmic averaging): in a previous paper I showed that achieving this in the context of the Liouville function would be equivalent to resolving the logarithmically averaged Sarnak conjecture, as well as establishing logarithmically averaged local Gowers uniformity of the Liouville function. However, in this paper we are able to avoid having to resolve these difficult conjectures to obtain partial results towards the (logarithmically averaged) Chowla and Elliott conjecture. For the Chowla conjecture, we can obtain all odd order correlations, in that
for all odd and all integers
(which, in the odd order case, are no longer required to be distinct). (Superficially, this looks like we have resolved “half of the cases” of the logarithmically averaged Chowla conjecture; but it seems the odd order correlations are significantly easier than the even order ones. For instance, because of the Katai-Bourgain-Sarnak-Ziegler criterion, one can basically deduce the odd order cases of (2) from the even order cases (after allowing for some dilations in the argument
).
For the more general Elliott conjecture, we can show that
for any , any integers
and any bounded multiplicative functions
, unless the product
weakly pretends to be a Dirichlet character
in the sense that
This can be seen to imply (2) as a special case. Even when does pretend to be a Dirichlet character
, we can still say something: if the limits
exist for each (which can be guaranteed if we pass to a suitable subsequence), then
is the uniform limit of periodic functions
, each of which is
–isotypic in the sense that
whenever
are integers with
coprime to the periods of
and
. This does not pin down the value of any single correlation
, but does put significant constraints on how these correlations may vary with
.
Among other things, this allows us to show that all possible length four sign patterns
of the Liouville function occur with positive density, and all
possible length four sign patterns
occur with the conjectured logarithmic density. (In a previous paper with Matomaki and Radziwill, we obtained comparable results for length three patterns of Liouville and length two patterns of Möbius.)
To describe the argument, let us focus for simplicity on the case of the Liouville correlations
assuming for sake of discussion that all limits exist. (In the paper, we instead use the device of generalised limits, as discussed in this previous post.) The idea is to combine together two rather different ways to control this function . The first proceeds by the entropy decrement method mentioned earlier, which roughly speaking works as follows. Firstly, we pick a prime
and observe that
for any
, which allows us to rewrite (3) as
Making the change of variables , we obtain
The difference between and
is negligible in the limit (here is where we crucially rely on the log-averaging), hence
and thus by (3) we have
The entropy decrement argument can be used to show that the latter limit is small for most (roughly speaking, this is because the factors
behave like independent random variables as
varies, so that concentration of measure results such as Hoeffding’s inequality can apply, after using entropy inequalities to decouple somewhat these random variables from the
factors). We thus obtain the approximate isotopy property
for most and
.
On the other hand, by the Furstenberg correspondence principle (as discussed in these previous posts), it is possible to express as a multiple correlation
for some probability space equipped with a measure-preserving invertible map
. Using results of Bergelson-Host-Kra, Leibman, and Le, this allows us to obtain a decomposition of the form
where is a nilsequence, and
goes to zero in density (even along the primes, or constant multiples of the primes). The original work of Bergelson-Host-Kra required ergodicity on
, which is very definitely a hypothesis that is not available here; however, the later work of Leibman removed this hypothesis, and the work of Le refined the control on
so that one still has good control when restricting to primes, or constant multiples of primes.
Ignoring the small error , we can now combine (5) to conclude that
Using the equidistribution theory of nilsequences (as developed in this previous paper of Ben Green and myself), one can break up further into a periodic piece
and an “irrational” or “minor arc” piece
. The contribution of the minor arc piece
can be shown to mostly cancel itself out after dilating by primes
and averaging, thanks to Vinogradov-type bilinear sum estimates (transferred to the primes). So we end up with
which already shows (heuristically, at least) the claim that can be approximated by periodic functions
which are isotopic in the sense that
But if is odd, one can use Dirichlet’s theorem on primes in arithmetic progressions to restrict to primes
that are
modulo the period of
, and conclude now that
vanishes identically, which (heuristically, at least) gives (2).
The same sort of argument works to give the more general bounds on correlations of bounded multiplicative functions. But for the specific task of proving (2), we initially used a slightly different argument that avoids using the ergodic theory machinery of Bergelson-Host-Kra, Leibman, and Le, but replaces it instead with the Gowers uniformity norm theory used to count linear equations in primes. Basically, by averaging (4) in using the “
-trick”, as well as known facts about the Gowers uniformity of the von Mangoldt function, one can obtain an approximation of the form
where ranges over a large range of integers coprime to some primorial
. On the other hand, by iterating (4) we have
for most semiprimes , and by again averaging over semiprimes one can obtain an approximation of the form
For odd, one can combine the two approximations to conclude that
. (This argument is not given in the current paper, but we plan to detail it in a subsequent one.)
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