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Over on the polymath blog, I’ve posted (on behalf of Dinesh Thakur) a new polymath proposal, which is to explain some numerically observed identities involving the irreducible polynomials P in the polynomial ring {\bf F}_2[t] over the finite field of characteristic two, the simplest of which is

\displaystyle \sum_P \frac{1}{1+P} = 0

(expanded in terms of Taylor series in u = 1/t).  Comments on the problem should be placed in the polymath blog post; if there is enough interest, we can start a formal polymath project on it.

Let {\lambda} denote the Liouville function. The prime number theorem is equivalent to the estimate

\displaystyle \sum_{n \leq x} \lambda(n) = o(x)

as {x \rightarrow \infty}, that is to say that {\lambda} exhibits cancellation on large intervals such as {[1,x]}. This result can be improved to give cancellation on shorter intervals. For instance, using the known zero density estimates for the Riemann zeta function, one can establish that

\displaystyle \int_X^{2X} |\sum_{x \leq n \leq x+H} \lambda(n)|\ dx = o( HX ) \ \ \ \ \ (1)

 

as {X \rightarrow \infty} if {X^{1/6+\varepsilon} \leq H \leq X} for some fixed {\varepsilon>0}; I believe this result is due to Ramachandra (see also Exercise 21 of this previous blog post), and in fact one could obtain a better error term on the right-hand side that for instance gained an arbitrary power of {\log X}. On the Riemann hypothesis (or the weaker density hypothesis), it was known that the {X^{1/6+\varepsilon}} could be lowered to {X^\varepsilon}.

Early this year, there was a major breakthrough by Matomaki and Radziwill, who (among other things) showed that the asymptotic (1) was in fact valid for any {H = H(X)} with {H \leq X} that went to infinity as {X \rightarrow \infty}, thus yielding cancellation on extremely short intervals. This has many further applications; for instance, this estimate, or more precisely its extension to other “non-pretentious” bounded multiplicative functions, was a key ingredient in my recent solution of the Erdös discrepancy problem, as well as in obtaining logarithmically averaged cases of Chowla’s conjecture, such as

\displaystyle \sum_{n \leq x} \frac{\lambda(n) \lambda(n+1)}{n} = o(\log x). \ \ \ \ \ (2)

 

It is of interest to twist the above estimates by phases such as the linear phase {n \mapsto e(\alpha n) := e^{2\pi i \alpha n}}. In 1937, Davenport showed that

\displaystyle \sup_\alpha |\sum_{n \leq x} \lambda(n) e(\alpha n)| \ll_A x \log^{-A} x

which of course improves the prime number theorem. Recently with Matomaki and Radziwill, we obtained a common generalisation of this estimate with (1), showing that

\displaystyle \sup_\alpha \int_X^{2X} |\sum_{x \leq n \leq x+H} \lambda(n) e(\alpha n)|\ dx = o(HX) \ \ \ \ \ (3)

 

as {X \rightarrow \infty}, for any {H = H(X) \leq X} that went to infinity as {X \rightarrow \infty}. We were able to use this estimate to obtain an averaged form of Chowla’s conjecture.

In that paper, we asked whether one could improve this estimate further by moving the supremum inside the integral, that is to say to establish the bound

\displaystyle \int_X^{2X} \sup_\alpha |\sum_{x \leq n \leq x+H} \lambda(n) e(\alpha n)|\ dx = o(HX) \ \ \ \ \ (4)

 

as {X \rightarrow \infty}, for any {H = H(X) \leq X} that went to infinity as {X \rightarrow \infty}. This bound is asserting that {\lambda} is locally Fourier-uniform on most short intervals; it can be written equivalently in terms of the “local Gowers {U^2} norm” as

\displaystyle \int_X^{2X} \sum_{1 \leq a \leq H} |\sum_{x \leq n \leq x+H} \lambda(n) \lambda(n+a)|^2\ dx = o( H^3 X )

from which one can see that this is another averaged form of Chowla’s conjecture (stronger than the one I was able to prove with Matomaki and Radziwill, but a consequence of the unaveraged Chowla conjecture). If one inserted such a bound into the machinery I used to solve the Erdös discrepancy problem, it should lead to further averaged cases of Chowla’s conjecture, such as

\displaystyle \sum_{n \leq x} \frac{\lambda(n) \lambda(n+1) \lambda(n+2)}{n} = o(\log x), \ \ \ \ \ (5)

 

though I have not fully checked the details of this implication. It should also have a number of new implications for sign patterns of the Liouville function, though we have not explored these in detail yet.

One can write (4) equivalently in the form

\displaystyle \int_X^{2X} \sum_{x \leq n \leq x+H} \lambda(n) e( \alpha(x) n + \beta(x) )\ dx = o(HX) \ \ \ \ \ (6)

 

uniformly for all {x}-dependent phases {\alpha(x), \beta(x)}. In contrast, (3) is equivalent to the subcase of (6) when the linear phase coefficient {\alpha(x)} is independent of {x}. This dependency of {\alpha(x)} on {x} seems to necessitate some highly nontrivial additive combinatorial analysis of the function {x \mapsto \alpha(x)} in order to establish (4) when {H} is small. To date, this analysis has proven to be elusive, but I would like to record what one can do with more classical methods like Vaughan’s identity, namely:

Proposition 1 The estimate (4) (or equivalently (6)) holds in the range {X^{2/3+\varepsilon} \leq H \leq X} for any fixed {\varepsilon>0}. (In fact one can improve the right-hand side by an arbitrary power of {\log X} in this case.)

The values of {H} in this range are far too large to yield implications such as new cases of the Chowla conjecture, but it appears that the {2/3} exponent is the limit of “classical” methods (at least as far as I was able to apply them), in the sense that one does not do any combinatorial analysis on the function {x \mapsto \alpha(x)}, nor does one use modern equidistribution results on “Type III sums” that require deep estimates on Kloosterman-type sums. The latter may shave a little bit off of the {2/3} exponent, but I don’t see how one would ever hope to go below {1/2} without doing some non-trivial combinatorics on the function {x \mapsto \alpha(x)}. UPDATE: I have come across this paper of Zhan which uses mean-value theorems for L-functions to lower the {2/3} exponent to {5/8}.

Let me now sketch the proof of the proposition, omitting many of the technical details. We first remark that known estimates on sums of the Liouville function (or similar functions such as the von Mangoldt function) in short arithmetic progressions, based on zero-density estimates for Dirichlet {L}-functions, can handle the “major arc” case of (4) (or (6)) where {\alpha} is restricted to be of the form {\alpha = \frac{a}{q} + O( X^{-1/6-\varepsilon} )} for {q = O(\log^{O(1)} X)} (the exponent here being of the same numerology as the {X^{1/6+\varepsilon}} exponent in the classical result of Ramachandra, tied to the best zero density estimates currently available); for instance a modification of the arguments in this recent paper of Koukoulopoulos would suffice. Thus we can restrict attention to “minor arc” values of {\alpha} (or {\alpha(x)}, using the interpretation of (6)).

Next, one breaks up {\lambda} (or the closely related Möbius function) into Dirichlet convolutions using one of the standard identities (e.g. Vaughan’s identity or Heath-Brown’s identity), as discussed for instance in this previous post (which is focused more on the von Mangoldt function, but analogous identities exist for the Liouville and Möbius functions). The exact choice of identity is not terribly important, but the upshot is that {\lambda(n)} can be decomposed into {\log^{O(1)} X} terms, each of which is either of the “Type I” form

\displaystyle \sum_{d \sim D; m \sim M: dm=n} a_d

for some coefficients {a_d} that are roughly of logarithmic size on the average, and scales {D, M} with {D \ll X^{2/3}} and {DM \sim X}, or else of the “Type II” form

\displaystyle \sum_{d \sim D; m \sim M: dm=n} a_d b_m

for some coefficients {a_d, b_m} that are roughly of logarithmic size on the average, and scales {D,M} with {X^{1/3} \ll D,M \ll X^{2/3}} and {DM \sim X}. As discussed in the previous post, the {2/3} exponent is a natural barrier in these identities if one is unwilling to also consider “Type III” type terms which are roughly of the shape of the third divisor function {\tau_3(n) := \sum_{d_1d_2d_3=1} 1}.

A Type I sum makes a contribution to { \sum_{x \leq n \leq x+H} \lambda(n) e( \alpha(x) n + \beta(x) )} that can be bounded (via Cauchy-Schwarz) in terms of an expression such as

\displaystyle \sum_{d \sim D} | \sum_{x/d \leq m \leq x/d+H/d} e(\alpha(x) dm )|^2.

The inner sum exhibits a lot of cancellation unless {\alpha(x) d} is within {O(D/H)} of an integer. (Here, “a lot” should be loosely interpreted as “gaining many powers of {\log X} over the trivial bound”.) Since {H} is significantly larger than {D}, standard Vinogradov-type manipulations (see e.g. Lemma 13 of these previous notes) show that this bad case occurs for many {d} only when {\alpha} is “major arc”, which is the case we have specifically excluded. This lets us dispose of the Type I contributions.

A Type II sum makes a contribution to { \sum_{x \leq n \leq x+H} \lambda(n) e( \alpha(x) n + \beta(x) )} roughly of the form

\displaystyle \sum_{d \sim D} | \sum_{x/d \leq m \leq x/d+H/d} b_m e(\alpha(x) dm)|.

We can break this up into a number of sums roughly of the form

\displaystyle \sum_{d = d_0 + O( H / M )} | \sum_{x/d_0 \leq m \leq x/d_0 + H/D} b_m e(\alpha(x) dm)|

for {d_0 \sim D}; note that the {d} range is non-trivial because {H} is much larger than {M}. Applying the usual bilinear sum Cauchy-Schwarz methods (e.g. Theorem 14 of these notes) we conclude that there is a lot of cancellation unless one has {\alpha(x) = a/q + O( \frac{X \log^{O(1)} X}{H^2} )} for some {q = O(\log^{O(1)} X)}. But with {H \geq X^{2/3+\varepsilon}}, {X \log^{O(1)} X/H^2} is well below the threshold {X^{-1/6-\varepsilon}} for the definition of major arc, so we can exclude this case and obtain the required cancellation.

The Chowla conjecture asserts that all non-trivial correlations of the Liouville function are asymptotically negligible; for instance, it asserts that

\displaystyle  \sum_{n \leq X} \lambda(n) \lambda(n+h) = o(X)

as {X \rightarrow \infty} for any fixed natural number {h}. This conjecture remains open, though there are a number of partial results (e.g. these two previous results of Matomaki, Radziwill, and myself).

A natural generalisation of Chowla’s conjecture was proposed by Elliott. For simplicity we will only consider Elliott’s conjecture for the pair correlations

\displaystyle  \sum_{n \leq X} g(n) \overline{g}(n+h).

For such correlations, the conjecture was that one had

\displaystyle  \sum_{n \leq X} g(n) \overline{g}(n+h) = o(X) \ \ \ \ \ (1)

as {X \rightarrow \infty} for any natural number {h}, as long as {g} was a completely multiplicative function with magnitude bounded by {1}, and such that

\displaystyle  \sum_p \hbox{Re} \frac{1 - g(p) \overline{\chi(p)} p^{-it}}{p} = +\infty \ \ \ \ \ (2)

for any Dirichlet character {\chi} and any real number {t}. In the language of “pretentious number theory”, as developed by Granville and Soundararajan, the hypothesis (2) asserts that the completely multiplicative function {g} does not “pretend” to be like the completely multiplicative function {n \mapsto \chi(n) n^{it}} for any character {\chi} and real number {t}. A condition of this form is necessary; for instance, if {g(n)} is precisely equal to {\chi(n) n^{it}} and {\chi} has period {q}, then {g(n) \overline{g}(n+q)} is equal to {1_{(n,q)=1} + o(1)} as {n \rightarrow \infty} and (1) clearly fails. The prime number theorem in arithmetic progressions implies that the Liouville function obeys (2), and so the Elliott conjecture contains the Chowla conjecture as a special case.

As it turns out, Elliott’s conjecture is false as stated, with the counterexample {g} having the property that {g} “pretends” locally to be the function {n \mapsto n^{it_j}} for {n} in various intervals {[1, X_j]}, where {X_j} and {t_j} go to infinity in a certain prescribed sense. See this paper of Matomaki, Radziwill, and myself for details. However, we view this as a technicality, and continue to believe that certain “repaired” versions of Elliott’s conjecture still hold. For instance, our counterexample does not apply when {g} is restricted to be real-valued rather than complex, and we believe that Elliott’s conjecture is valid in this setting. Returning to the complex-valued case, we still expect the asymptotic (1) provided that the condition (2) is replaced by the stronger condition

\displaystyle  \sup_{|t| \leq X} |\sum_{p \leq X} \hbox{Re} \frac{1 - g(p) \overline{\chi(p)} p^{-it}}{p}| \rightarrow +\infty

as {X \rightarrow +\infty} for all fixed Dirichlet characters {\chi}. In our paper we supported this claim by establishing a certain “averaged” version of this conjecture; see that paper for further details. (See also this recent paper of Frantzikinakis and Host which establishes a different averaged version of this conjecture.)

One can make a stronger “non-asymptotic” version of this corrected Elliott conjecture, in which the {X} parameter does not go to infinity, or equivalently that the function {g} is permitted to depend on {X}:

Conjecture 1 (Non-asymptotic Elliott conjecture) Let {\varepsilon > 0}, let {A \geq 1} be sufficiently large depending on {\varepsilon}, and let {X} be sufficiently large depending on {A,\varepsilon}. Suppose that {g} is a completely multiplicative function with magnitude bounded by {1}, such that

\displaystyle  \inf_{|t| \leq AX} |\sum_{p \leq X} \hbox{Re} \frac{1 - g(p) \overline{\chi(p)} p^{-it}}{p}| \geq A

for all Dirichlet characters {\chi} of period at most {A}. Then one has

\displaystyle  |\sum_{n \leq X} g(n) \overline{g(n+h)}| \leq \varepsilon X

for all natural numbers {1 \leq h \leq 1/\varepsilon}.

The {\varepsilon}-dependent factor {A} in the constraint {|t| \leq AX} is necessary, as can be seen by considering the completely multiplicative function {g(n) := n^{2iX}} (for instance). Again, the results in my previous paper with Matomaki and Radziwill can be viewed as establishing an averaged version of this conjecture.

Meanwhile, we have the following conjecture that is the focus of the Polymath5 project:

Conjecture 2 (Erdös discrepancy conjecture) For any function {f: {\bf N} \rightarrow \{-1,+1\}}, the discrepancy

\displaystyle  \sup_{n,d \in {\bf N}} |\sum_{j=1}^n f(jd)|

is infinite.

It is instructive to compute some near-counterexamples to Conjecture 2 that illustrate the difficulty of the Erdös discrepancy problem. The first near-counterexample is that of a non-principal Dirichlet character {f(n) = \chi(n)} that takes values in {\{-1,0,+1\}} rather than {\{-1,+1\}}. For this function, one has from the complete multiplicativity of {\chi} that

\displaystyle  |\sum_{j=1}^n f(jd)| = |\sum_{j=1}^n \chi(j) \chi(d)|

\displaystyle  \leq |\sum_{j=1}^n \chi(j)|.

If {q} denotes the period of {\chi}, then {\chi} has mean zero on every interval of length {q}, and thus

\displaystyle  |\sum_{j=1}^n f(jd)| \leq |\sum_{j=1}^n \chi(j)| \leq q.

Thus {\chi} has bounded discrepancy.

Of course, this is not a true counterexample to Conjecture 2 because {\chi} can take the value {0}. Let us now consider the following variant example, which is the simplest member of a family of examples studied by Borwein, Choi, and Coons. Let {\chi = \chi_3} be the non-principal Dirichlet character of period {3} (thus {\chi(n)} equals {+1} when {n=1 \hbox{ mod } 3}, {-1} when {n = 2 \hbox{ mod } 3}, and {0} when {n = 0 \hbox{ mod } 3}), and define the completely multiplicative function {f = \tilde \chi: {\bf N} \rightarrow \{-1,+1\}} by setting {\tilde \chi(p) := \chi(p)} when {p \neq 3} and {\tilde \chi(3) = +1}. This is about the simplest modification one can make to the previous near-counterexample to eliminate the zeroes. Now consider the sum

\displaystyle  \sum_{j=1}^n \tilde \chi(j)

with {n := 1 + 3 + 3^2 + \dots + 3^k} for some large {k}. Writing {j = 3^a m} with {m} coprime to {3} and {a} at most {k}, we can write this sum as

\displaystyle  \sum_{a=0}^k \sum_{1 \leq m \leq n/3^j} \tilde \chi(3^a m).

Now observe that {\tilde \chi(3^a m) = \tilde \chi(3)^a \tilde \chi(m) = \chi(m)}. The function {\chi} has mean zero on every interval of length three, and {\lfloor n/3^j\rfloor} is equal to {1} mod {3}, and thus

\displaystyle  \sum_{1 \leq m \leq n/3^j} \tilde \chi(3^a m) = 1

for every {a=0,\dots,k}, and thus

\displaystyle  \sum_{j=1}^n \tilde \chi(j) = k+1 \gg \log n.

Thus {\tilde \chi} also has unbounded discrepancy, but only barely so (it grows logarithmically in {n}). These examples suggest that the main “enemy” to proving Conjecture 2 comes from completely multiplicative functions {f} that somehow “pretend” to be like a Dirichlet character but do not vanish at the zeroes of that character. (Indeed, the special case of Conjecture 2 when {f} is completely multiplicative is already open, appears to be an important subcase.)

All of these conjectures remain open. However, I would like to record in this blog post the following striking connection, illustrating the power of the Elliott conjecture (particularly in its nonasymptotic formulation):

Theorem 3 (Elliott conjecture implies unbounded discrepancy) Conjecture 1 implies Conjecture 2.

The argument relies heavily on two observations that were previously made in connection with the Polymath5 project. The first is a Fourier-analytic reduction that replaces the Erdos Discrepancy Problem with an averaged version for completely multiplicative functions {g}. An application of Cauchy-Schwarz then shows that any counterexample to that version will violate the conclusion of Conjecture 1, so if one assumes that conjecture then {g} must pretend to be like a function of the form {n \mapsto \chi(n) n^{it}}. One then uses (a generalisation) of a second argument from Polymath5 to rule out this case, basically by reducing matters to a more complicated version of the Borwein-Choi-Coons analysis. Details are provided below the fold.

There is some hope that the Chowla and Elliott conjectures can be attacked, as the parity barrier which is so impervious to attack for the twin prime conjecture seems to be more permeable in this setting. (For instance, in my previous post I raised a possible approach, based on establishing expander properties of a certain random graph, which seems to get around the parity problem, in principle at least.)

(Update, Sep 25: fixed some treatment of error terms, following a suggestion of Andrew Granville.)

Read the rest of this entry »

Here’s a cute identity I discovered by accident recently. Observe that

\displaystyle  \frac{d}{dx} (1+x^2)^{0/2} = 0

\displaystyle  \frac{d^2}{dx^2} (1+x^2)^{1/2} = \frac{1}{(1+x^2)^{3/2}}

\displaystyle  \frac{d^3}{dx^3} (1+x^2)^{2/2} = 0

\displaystyle  \frac{d^4}{dx^4} (1+x^2)^{3/2} = \frac{9}{(1+x^2)^{5/2}}

\displaystyle  \frac{d^5}{dx^5} (1+x^2)^{4/2} = 0

\displaystyle  \frac{d^6}{dx^6} (1+x^2)^{5/2} = \frac{225}{(1+x^2)^{7/2}}

and so one can conjecture that one has

\displaystyle  \frac{d^{k+1}}{dx^{k+1}} (1+x^2)^{k/2} = 0

when k is even, and

\displaystyle  \frac{d^{k+1}}{dx^{k+1}} (1+x^2)^{k/2} = \frac{(1 \times 3 \times \dots \times k)^2}{(1+x^2)^{(k+2)/2}}

when k is odd. This is obvious in the even case since (1+x^2)^{k/2} is a polynomial of degree k, but I struggled for a while with the odd case before finding a slick three-line proof. (I was first trying to prove the weaker statement that \frac{d^{k+1}}{dx^{k+1}} (1+x^2)^{k/2} was non-negative, but for some strange reason I was only able to establish this by working out the derivative exactly, rather than by using more analytic methods, such as convexity arguments.) I thought other readers might like the challenge (and also I’d like to see some other proofs), so rather than post my own proof immediately, I’ll see if anyone would like to supply their own proofs or thoughts in the comments. Also I am curious to know if this identity is connected to any other existing piece of mathematics.

The lonely runner conjecture is the following open problem:

Conjecture 1 Suppose one has {n \geq 1} runners on the unit circle {{\bf R}/{\bf Z}}, all starting at the origin and moving at different speeds. Then for each runner, there is at least one time {t} for which that runner is “lonely” in the sense that it is separated by a distance at least {1/n} from all other runners.

One can normalise the speed of the lonely runner to be zero, at which point the conjecture can be reformulated (after replacing {n} by {n+1}) as follows:

Conjecture 2 Let {v_1,\dots,v_n} be non-zero real numbers for some {n \geq 1}. Then there exists a real number {t} such that the numbers {tv_1,\dots,tv_n} are all a distance at least {\frac{1}{n+1}} from the integers, thus {\|tv_1\|_{{\bf R}/{\bf Z}},\dots,\|tv_n\|_{{\bf R}/{\bf Z}} \geq \frac{1}{n+1}} where {\|x\|_{{\bf R}/{\bf Z}}} denotes the distance of {x} to the nearest integer.

This conjecture has been proven for {n \leq 7}, but remains open for larger {n}. The bound {\frac{1}{n+1}} is optimal, as can be seen by looking at the case {v_i=i} and applying the Dirichlet approximation theorem. Note that for each non-zero {v}, the set {\{ t \in {\bf R}: \|vt\|_{{\bf R}/{\bf Z}} \leq r \}} has (Banach) density {2r} for any {0 < r < 1/2}, and from this and the union bound we can easily find {t \in {\bf R}} for which

\displaystyle \|tv_1\|_{{\bf R}/{\bf Z}},\dots,\|tv_n\|_{{\bf R}/{\bf Z}} \geq \frac{1}{2n}-\varepsilon

for any {\varepsilon>0}, but it has proven to be quite challenging to remove the factor of {2} to increase {\frac{1}{2n}} to {\frac{1}{n+1}}. (As far as I know, even improving {\frac{1}{2n}} to {\frac{1+c}{2n}} for some absolute constant {c>0} and sufficiently large {n} remains open.)

The speeds {v_1,\dots,v_n} in the above conjecture are arbitrary non-zero reals, but it has been known for some time that one can reduce without loss of generality to the case when the {v_1,\dots,v_n} are rationals, or equivalently (by scaling) to the case where they are integers; see e.g. Section 4 of this paper of Bohman, Holzman, and Kleitman.

In this post I would like to remark on a slight refinement of this reduction, in which the speeds {v_1,\dots,v_n} are integers of bounded size, where the bound depends on {n}. More precisely:

Proposition 3 In order to prove the lonely runner conjecture, it suffices to do so under the additional assumption that the {v_1,\dots,v_n} are integers of size at most {n^{Cn^2}}, where {C} is an (explicitly computable) absolute constant. (More precisely: if this restricted version of the lonely runner conjecture is true for all {n \leq n_0}, then the original version of the conjecture is also true for all {n \leq n_0}.)

In principle, this proposition allows one to verify the lonely runner conjecture for a given {n} in finite time; however the number of cases to check with this proposition grows faster than exponentially in {n}, and so this is unfortunately not a feasible approach to verifying the lonely runner conjecture for more values of {n} than currently known.

One of the key tools needed to prove this proposition is the following additive combinatorics result. Recall that a generalised arithmetic progression (or {GAP}) in the reals {{\bf R}} is a set of the form

\displaystyle  P = \{ n_1 v_1 + \dots + n_d v_d: n_1,\dots,n_d \in {\bf Z}; |n_1| \leq N_1, \dots, |n_d| \leq N_d \}

for some {v_1,\dots,v_d \in {\bf R}} and {N_1,\dots,N_d > 0}; the quantity {d} is called the rank of the progression. If {t>0}, the progression {P} is said to be {t}-proper if the sums {n_1 v_1 + \dots + n_d v_d} with {|n_i| \leq t N_i} for {i=1,\dots,d} are all distinct. We have

Lemma 4 (Progressions lie inside proper progressions) Let {P} be a GAP of rank {d} in the reals, and let {t \geq 1}. Then {P} is contained in a {t}-proper GAP {Q} of rank at most {d}, with

\displaystyle |Q| \leq (2t)^d d^{6d^2} \prod_{i=1}^d (2N_i+1).

Proof: See Theorem 2.1 of this paper of Bilu. (Very similar results can also be found in Theorem 3.40 of my book with Van Vu, or Theorem 1.10 of this paper of mine with Van Vu.) \Box

Now let {n \geq 1}, and assume inductively that the lonely runner conjecture has been proven for all smaller values of {n}, as well as for the current value of {n} in the case that {v_1,\dots,v_n} are integers of size at most {n^{Cn^2}} for some sufficiently large {C}. We will show that the lonely runner conjecture holds in general for this choice of {n}.

let {v_1,\dots,v_n} be non-zero real numbers. Let {C_0} be a large absolute constant to be chosen later. From the above lemma applied to the GAP {\{ n_1 v_1 + \dots + n_d v_d: n_1,\dots,n_d \in \{-1,0,1\}\}}, one can find a {n^{C_0n}}-proper GAP {Q} of rank at most {n} containing {\{v_1,\dots,v_n\}} such that

\displaystyle  |Q| \leq (6n^{C_0 n})^n n^{6n^2};

in particular {|Q| \leq n^{Cn^2}} if {C} is large enough depending on {C_0}.

We write

\displaystyle  Q = \{ n_1 w_1 + \dots + n_d w_d: n_1,\dots,n_d \in {\bf Z}; |n_1| \leq N_1,\dots,|n_d| \leq N_d \}

for some {d \leq n}, {w_1,\dots,w_d}, and {N_1,\dots,N_d \geq 0}. We thus have {v_i = \phi(a_i)} for {i=1,\dots,n}, where {\phi: {\bf R}^d \rightarrow {\bf R}} is the linear map {\phi(n_1,\dots,n_d) := n_1 w_1 + \dots + n_d w_d} and {a_1,\dots,a_n \in {\bf Z}^d} are non-zero and lie in the box {\{ (n_1,\dots,n_d) \in {\bf R}^d: |n_1| \leq N_1,\dots,|n_d| \leq N_d \}}.

We now need an elementary lemma that allows us to create a “collision” between two of the {a_1,\dots,a_n} via a linear projection, without making any of the {a_i} collide with the origin:

Lemma 5 Let {a_1,\dots,a_n \in {\bf R}^d} be non-zero vectors that are not all collinear with the origin. Then, after replacing one or more of the {a_i} with their negatives {-a_i} if necessary, there exists a pair {a_i,a_j} such that {a_i-a_j \neq 0}, and such that none of the {a_1,\dots,a_n} is a scalar multiple of {a_i-a_j}.

Proof: We may assume that {d \geq 2}, since the {d \leq 1} case is vacuous. Applying a generic linear projection to {{\bf R}^2} (which does not affect collinearity, or the property that a given {a_k} is a scalar multiple of {a_i-a_j}), we may then reduce to the case {d=2}.

By a rotation and relabeling, we may assume that {a_1} lies on the negative {x}-axis; by flipping signs as necessary we may then assume that all of the {a_2,\dots,a_n} lie in the closed right half-plane. As the {a_i} are not all collinear with the origin, one of the {a_i} lies off of the {x}-axis, by relabeling, we may assume that {a_2} lies off of the {x} axis and makes a minimal angle with the {x}-axis. Then the angle of {a_2-a_1} with the {x}-axis is non-zero but smaller than any non-zero angle that any of the {a_i} make with this axis, and so none of the {a_i} are a scalar multiple of {a_2-a_1}, and the claim follows. \Box

We now return to the proof of the proposition. If the {a_1,\dots,a_n} are all collinear with the origin, then {\phi(a_1),\dots,\phi(a_n)} lie in a one-dimensional arithmetic progression {\{ mv: |m| \leq |Q| \}}, and then by rescaling we may take the {v_1,\dots,v_n} to be integers of magnitude at most {|Q| \leq n^{Cn}}, at which point we are done by hypothesis. Thus, we may assume that the {a_1,\dots,a_n} are not all collinear with the origin, and so by the above lemma and relabeling we may assume that {a_n-a_1} is non-zero, and that none of the {a_i} are scalar multiples of {a_n-a_1}.

We write

\displaystyle  a_n-a_1 = (c_1,\dots,c_d) \ \ \ \ \ (1)

with {|c_i| \leq 2 N_i} for {i=1,\dots,d}; by relabeling we may assume without loss of generality that {c_d} is non-zero, and furthermore that

\displaystyle  \frac{|c_i|}{N_i} \leq \frac{|c_d|}{N_d}

for {i=1,\dots,d}. We can also factor

\displaystyle  (c_1,\dots,c_d) = q (c'_1,\dots,c'_d) \ \ \ \ \ (2)

where {q} is a natural number and {c'_1,\dots,c'_d} have no common factor.

We now define a variant {\tilde \phi: {\bf R}^d \rightarrow {\bf R}} of {\phi: {\bf R}^d \rightarrow {\bf R}} by the map

\displaystyle  \tilde \phi(n_1,\dots,n_d) := n_1 \tilde w_1 + \dots + n_{d-1} \tilde w_{d-1} - \frac{n_d}{c_d} (c_1 \tilde w_1 + \dots + c_{d-1} \tilde w_{d-1}),

where the {\tilde w_1,\dots,\tilde w_{d-1}} are real numbers that are linearly independent over {{\bf Q}}, whose precise value will not be of importance in our argument. This is a linear map with the property that {\tilde \phi(a_n-a_1)=0}, so that {\tilde \phi(a_1),\dots,\tilde \phi(a_n)} consists of at most {n-1} distinct real numbers, which are non-zero since none of the {a_i} are scalar multiples of {a_n-a_1}, and the {\tilde w_i} are linearly independent over {{\bf Q}}. As we are assuming inductively that the lonely runner conjecture holds for {n-1}, we conclude (after deleting duplicates) that there exists at least one real number {\tilde t} such that

\displaystyle  \| \tilde t \tilde \phi(a_1) \|_{{\bf R}/{\bf Z}}, \dots, \| \tilde t \tilde \phi(a_n) \|_{{\bf R}/{\bf Z}} \geq \frac{1}{n}.

We would like to “approximate” {\phi} by {\tilde \phi} to then conclude that there is at least one real number {t} such that

\displaystyle  \| t \phi(a_1) \|_{{\bf R}/{\bf Z}}, \dots, \| t \phi(a_n) \|_{{\bf R}/{\bf Z}} \geq \frac{1}{n+1}.

It turns out that we can do this by a Fourier-analytic argument taking advantage of the {n^{C_0 n}}-proper nature of {Q}. Firstly, we see from the Dirichlet approximation theorem that one has

\displaystyle  \| \tilde t \tilde \phi(a_1) \|_{{\bf R}/{\bf Z}}, \dots, \| \tilde t \tilde \phi(a_n) \|_{{\bf R}/{\bf Z}} \leq \frac{1}{10 n^2}

for a set {\tilde t} of reals of (Banach) density {\gg n^{-O(n)}}. Thus, by the triangle inequality, we have

\displaystyle  \| \tilde t \tilde \phi(a_1) \|_{{\bf R}/{\bf Z}}, \dots, \| \tilde t \tilde \phi(a_n) \|_{{\bf R}/{\bf Z}} \geq \frac{1}{n} - \frac{1}{10n^2}

for a set {\tilde t} of reals of density {\gg n^{-O(n)}}.

Applying a smooth Fourier multiplier of Littlewood-Paley type, one can find a trigonometric polynomial

\displaystyle  \eta(x) = \sum_{m: |m| \leq n^{C_0 n/10}} b_m e^{2\pi i mx}

which takes values in {[0,1]}, is {\gg 1} for {\|x\|_{{\bf R}/{\bf Z}} \geq \frac{1}{n} - \frac{1}{10n^2}}, and is no larger than {O( n^{-100 C_0n} )} for {\|x\|_{{\bf R}/{\bf Z}} \leq \frac{1}{n+1}}. We then have

\displaystyle  \mathop{\bf E}_t \prod_{j=1}^n \eta( t \tilde \phi(a_j) ) \gg n^{-O(n)}

where {\mathop{\bf E}_t f(t)} denotes the mean value of a quasiperiodic function {f} on the reals {{\bf R}}. We expand the left-hand side out as

\displaystyle  \sum_{m_1,\dots,m_n: m_1 \tilde \phi(a_1) + \dots + m_n \tilde \phi(a_n) = 0} b_{m_1} \dots b_{m_n}.

From the genericity of {\tilde w_1,\dots,\tilde w_n}, we see that the constraint

\displaystyle  m_1 \tilde \phi(a_1) + \dots + m_n \tilde \phi(a_n) = 0

occurs if and only if {m_1 a_1 + \dots + m_n a_n} is a scalar multiple of {a_n-a_1}, or equivalently (by (1), (2)) an integer multiple of {(c'_1,\dots,c'_d)}. Thus

\displaystyle  \sum_{m_1,\dots,m_n: m_1 a_1 + \dots + m_n a_n \in {\bf Z} (c'_1,\dots,c'_d)} b_{m_1} \dots b_{m_n} \gg n^{-O(n)}. \ \ \ \ \ (3)

Next, we consider the average

\displaystyle  \mathop{\bf E}_t \varphi( t \xi ) \prod_{j=1}^n \eta( t v_j ) \ \ \ \ \ (4)

where

\displaystyle  \xi := c'_1 w_1 + \dots + c'_d w_d. \ \ \ \ \ (5)

and {\varphi} is the Dirichlet series

\displaystyle  \varphi(x) := \sum_{m: |m| \leq n^{C_0 n/2}} e^{2\pi i mx}.

By Fourier expansion and writing {v_j = \phi(a_j)}, we may write (4) as

\displaystyle  \sum_{m,m_1,\dots,m_n: |m| \leq n^{C_0n/2}; m_1 \phi(a_1) + \dots + m_n \phi(a_n) = m \xi} b_{m_1} \dots b_{m_n}.

The support of the {b_{m_i}} implies that {|m_i| \leq n^{C_0n/10}}. Because of the {n^{C_0 n}}-properness of {Q}, we see (for {n} large enough) that the equation

\displaystyle  m_1 \phi(a_1) + \dots + m_n \phi(a_n) = m \xi \ \ \ \ \ (6)

implies that

\displaystyle  m_1 a_1 + \dots + m_n a_n \in {\bf Z} (c'_1,\dots,c'_d) \ \ \ \ \ (7)

and conversely that (7) implies that (6) holds for some {m} with {|m| \leq n^{C_0 n/2}}. From (3) we thus have

\displaystyle  \mathop{\bf E}_t \varphi( t \xi ) \prod_{j=1}^n \eta( t v_j ) \gg n^{-O(1)}.

In particular, there exists a {t} such that

\displaystyle  \varphi( t \xi ) \prod_{j=1}^n \eta( t v_j ) \gg n^{-O(1)}.

Since {\varphi} is bounded in magnitude by {n^{C_0n/2}}, and {\eta} is bounded by {1}, we thus have

\displaystyle  \eta(t v_j) \gg n^{-C_0 n/2 - O(1)}

for each {1 \leq j \leq n}, which by the size properties of {\eta} implies that {\|tv_j\|_{{\bf R}/{\bf Z}} \geq \frac{1}{n+1}} for all {1 \leq j \leq n}, giving the lonely runner conjecture for {n}.

The (presumably) final article arising from the Polymath8 project has now been uploaded to the arXiv as “The “bounded gaps between primes” Polymath project – a retrospective“.  This article, submitted to the Newsletter of the European Mathematical Society, consists of personal contributions from ten different participants (at varying levels of stage of career, and intensity of participation) on their own experiences with the project, and some thoughts as to what lessons to draw for any subsequent Polymath projects.  (At present, I do not know of any such projects being proposed, but from recent experience I would imagine that some opportunity suitable for a Polymath approach will present itself at some point in the near future.)

This post will also serve as the latest (and probably last) of the Polymath8 threads (rolling over this previous post), to wrap up any remaining discussion about any aspect of this project.

I’ve just uploaded to the arXiv the D.H.J. Polymath paper “Variants of the Selberg sieve, and bounded intervals containing many primes“, which is the second paper to be produced from the Polymath8 project (the first one being discussed here). We’ll refer to this latter paper here as the Polymath8b paper, and the former as the Polymath8a paper. As with Polymath8a, the Polymath8b paper is concerned with the smallest asymptotic prime gap

\displaystyle H_1 := \liminf_{n \rightarrow \infty}(p_{n+1}-p_n),

where {p_n} denotes the {n^{th}} prime, as well as the more general quantities

\displaystyle H_m := \liminf_{n \rightarrow \infty}(p_{n+m}-p_n).

In the breakthrough paper of Goldston, Pintz, and Yildirim, the bound {H_1 \leq 16} was obtained under the strong hypothesis of the Elliott-Halberstam conjecture. An unconditional bound on {H_1}, however, remained elusive until the celebrated work of Zhang last year, who showed that

\displaystyle H_1 \leq 70{,}000{,}000.

The Polymath8a paper then improved this to {H_1 \leq 4{,}680}. After that, Maynard introduced a new multidimensional Selberg sieve argument that gave the substantial improvement

\displaystyle H_1 \leq 600

unconditionally, and {H_1 \leq 12} on the Elliott-Halberstam conjecture; furthermore, bounds on {H_m} for higher {m} were obtained for the first time, and specifically that {H_m \ll m^3 e^{4m}} for all {m \geq 1}, with the improvements {H_2 \leq 600} and {H_m \ll m^3 e^{2m}} on the Elliott-Halberstam conjecture. (I had independently discovered the multidimensional sieve idea, although I did not obtain Maynard’s specific numerical results, and my asymptotic bounds were a bit weaker.)

In Polymath8b, we obtain some further improvements. Unconditionally, we have {H_1 \leq 246} and {H_m \ll m e^{(4 - \frac{28}{157}) m}}, together with some explicit bounds on {H_2,H_3,H_4,H_5}; on the Elliott-Halberstam conjecture we have {H_m \ll m e^{2m}} and some numerical improvements to the {H_2,H_3,H_4,H_5} bounds; and assuming the generalised Elliott-Halberstam conjecture we have the bound {H_1 \leq 6}, which is best possible from sieve-theoretic methods thanks to the parity problem obstruction.

There were a variety of methods used to establish these results. Maynard’s paper obtained a criterion for bounding {H_m} which reduced to finding a good solution to a certain multidimensional variational problem. When the dimension parameter {k} was relatively small (e.g. {k \leq 100}), we were able to obtain good numerical solutions both by continuing the method of Maynard (using a basis of symmetric polynomials), or by using a Krylov iteration scheme. For large {k}, we refined the asymptotics and obtained near-optimal solutions of the variational problem. For the {H_1} bounds, we extended the reach of the multidimensional Selberg sieve (particularly under the assumption of the generalised Elliott-Halberstam conjecture) by allowing the function {F} in the multidimensional variational problem to extend to a larger region of space than was previously admissible, albeit with some tricky new constraints on {F} (and penalties in the variational problem). This required some unusual sieve-theoretic manipulations, notably an “epsilon trick”, ultimately relying on the elementary inequality {(a+b)^2 \geq a^2 + 2ab}, that allowed one to get non-trivial lower bounds for sums such as {\sum_n (a(n)+b(n))^2} even if the sum {\sum_n b(n)^2} had no non-trivial estimates available; and a way to estimate divisor sums such as {\sum_{n\leq x} \sum_{d|n} \lambda_d} even if {d} was permitted to be comparable to or even exceed {x}, by using the fundamental theorem of arithmetic to factorise {n} (after restricting to the case when {n} is almost prime). I hope that these sieve-theoretic tricks will be useful in future work in the subject.

With this paper, the Polymath8 project is almost complete; there is still a little bit of scope to push our methods further and get some modest improvement for instance to the {H_1 \leq 246} bound, but this would require a substantial amount of effort, and it is probably best to instead wait for some new breakthrough in the subject to come along. One final task we are performing is to write up a retrospective article on both the 8a and 8b experiences, an incomplete writeup of which can be found here. If anyone wishes to contribute some commentary on these projects (whether you were an active contributor, an occasional contributor, or a silent “lurker” in the online discussion), please feel free to do so in the comments to this post.

This should be the final thread (for now, at least) for the Polymath8 project (encompassing the original Polymath8a paper, the nearly finished Polymath8b paper, and the retrospective paper), superseding the previous Polymath8b thread (which was quite full) and the Polymath8a/retrospective thread (which was more or less inactive).

On Polymath8a: I talked briefly with Andrew Granville, who is handling the paper for Algebra & Number Theory, and he said that a referee report should be coming in soon.  Apparently length of the paper is a bit of an issue (not surprising, as it is 163 pages long) and there will be some suggestions to trim the size down a bit.

In view of the length issue at A&NT, I’m now leaning towards taking up Ken Ono’s offer to submit the Polymath8b paper to the new open access journal “Research in the Mathematical Sciences“.  I think the paper is almost ready to be submitted (after the current participants sign off on it, of course), but it might be worth waiting on the Polymath8a referee report in case the changes suggested impact the 8b paper.

Finally, it is perhaps time to start working on the retrospective article, and collect some impressions from participants.  I wrote up a quick draft of my own experiences, and also pasted in Pace Nielsen’s thoughts, as well as a contribution from an undergraduate following the project (Andrew Gibson).  Hopefully we can collect a few more (either through comments on this blog, through email, or through Dropbox), and then start working on editing them together and finding some suitable concluding points to make about the Polymath8 project, and what lessons we can take from it for future projects of this type.

This is the eleventh thread for the Polymath8b project to obtain new bounds for the quantity

H_m := \liminf_{n \to\infty} p_{n+m} - p_n;

the previous thread may be found here.

The main focus is now on writing up the results, with a draft paper close to completion here (with the directory of source files here).    Most of the sections are now written up more or less completely, with the exception of the appendix on narrow admissible tuples, which was awaiting the bounds on such tuples to stabilise.  There is now also an acknowledgments section (linking to the corresponding page on the wiki, which participants should check to see if their affiliations etc. are posted correctly), and in the final remarks section there is now also some discussion about potential improvements to the H_m bounds.  I’ve also added some mention of a recent paper of Banks, Freiberg and Maynard which makes use of some of our results (in particular, that M_{50,1/25} > 4).  On the other hand, the portions of the writeup relating to potential improvements to the MPZ estimates have been commented out, as it appears that one cannot easily obtain the exponential sum estimates required to make those go through.  (Perhaps, if there are significant new developments, one could incorporate them into a putative Polymath8c project, although at present I think there’s not much urgency to start over once again.)

Regarding the numerics in Section 7 of the paper, one thing which is missing at present is some links to code in case future readers wish to verify the results; alternatively one could include such code and data into the arXiv submission.

It’s about time to discuss possible journals to submit the paper to.  Ken Ono has invited us to submit to his new journal, “Research in the Mathematical Sciences“.  Another option would be to submit to the same journal “Algebra & Number Theory” that is currently handling our Polymath8a paper (no news on the submission there, but it is a very long paper), although I think the papers are independent enough that it is not absolutely necessary to place them in the same journal.  A third natural choice is “Mathematics of Computation“, though I should note that when the Polymath4 paper was submitted there, the editors required us to use our real names instead of the D.H.J. Polymath pseudonym as it would have messed up their metadata system otherwise.  (But I can check with the editor there before submitting to see if there is some workaround now, perhaps their policies have changed.)  At present I have no strong preferences regarding journal selection, and would welcome further thoughts and proposals.  (It is perhaps best to avoid the journals that I am editor or associate editor of, namely Amer. J. Math, Forum of Mathematics, Analysis & PDE, and Dynamics and PDE, due to conflict of interest (and in the latter two cases, specialisation to a different area of mathematics)).

 

 

 

 

This is the tenth thread for the Polymath8b project to obtain new bounds for the quantity

H_m := \liminf_{n \to\infty} p_{n+m} - p_n;

the previous thread may be found here.

Numerical progress on these bounds have slowed in recent months, although we have very recently lowered the unconditional bound on H_1 from 252 to 246 (see the wiki page for more detailed results).  While there may still be scope for further improvement (particularly with respect to bounds for H_m with m=2,3,4,5, which we have not focused on for a while, it looks like we have reached the point of diminishing returns, and it is time to turn to the task of writing up the results.

A draft version of the paper so far may be found here (with the directory of source files here).  Currently, the introduction and the sieve-theoretic portions of the paper are written up, although the sieve-theoretic arguments are surprisingly lengthy, and some simplification (or other reorganisation) may well be possible.  Other portions of the paper that have not yet been written up include the asymptotic analysis of M_k for large k (leading in particular to results for m=2,3,4,5), and a description of the quadratic programming that is used to estimate M_k for small and medium k.  Also we will eventually need an appendix to summarise the material from Polymath8a that we would use to generate various narrow admissible tuples.

One issue here is that our current unconditional bounds on H_m for m=2,3,4,5 rely on a distributional estimate on the primes which we believed to be true in Polymath8a, but never actually worked out (among other things, there was some delicate algebraic geometry issues concerning the vanishing of certain cohomology groups that was never resolved).  This issue does not affect the m=1 calculations, which only use the Bombieri-Vinogradov theorem or else assume the generalised Elliott-Halberstam conjecture.  As such, we will have to rework the computations for these H_m, given that the task of trying to attain the conjectured distributional estimate on the primes would be a significant amount of work that is rather disjoint from the rest of the Polymath8b writeup.  One could simply dust off the old maple code for this (e.g. one could tweak the code here, with the constraint  1080*varpi/13+ 330*delta/13<1  being replaced by 600*varpi/7+180*delta/7<1), but there is also a chance that our asymptotic bounds for M_k (currently given in messy detail here) could be sharpened.  I plan to look at this issue fairly soon.

Also, there are a number of smaller observations (e.g. the parity problem barrier that prevents us from ever getting a better bound on H_1 than 6) that should also go into the paper at some point; the current outline of the paper as given in the draft is not necessarily comprehensive.

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