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Let denote the Liouville function. The prime number theorem is equivalent to the estimate

as , that is to say that exhibits cancellation on large intervals such as . 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

as if for some fixed ; 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 . On the Riemann hypothesis (or the weaker density hypothesis), it was known that the could be lowered to .

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* with that went to infinity as , 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

It is of interest to twist the above estimates by phases such as the linear phase . In 1937, Davenport showed that

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

as , for any that went to infinity as . 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

as , for any that went to infinity as . This bound is asserting that is locally Fourier-uniform on most short intervals; it can be written equivalently in terms of the “local Gowers norm” as

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

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

uniformly for all -dependent phases . In contrast, (3) is equivalent to the subcase of (6) when the linear phase coefficient is independent of . This dependency of on seems to necessitate some highly nontrivial additive combinatorial analysis of the function in order to establish (4) when 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 1The estimate (4) (or equivalently (6)) holds in the range for any fixed . (In fact one can improve the right-hand side by an arbitrary power of in this case.)

The values of in this range are far too large to yield implications such as new cases of the Chowla conjecture, but it appears that the 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 , 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 exponent, but I don’t see how one would ever hope to go below without doing some non-trivial combinatorics on the function . UPDATE: I have come across this paper of Zhan which uses mean-value theorems for L-functions to lower the exponent to .

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 -functions, can handle the “major arc” case of (4) (or (6)) where is restricted to be of the form for (the exponent here being of the same numerology as the 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 (or , using the interpretation of (6)).

Next, one breaks up (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 can be decomposed into terms, each of which is either of the “Type I” form

for some coefficients that are roughly of logarithmic size on the average, and scales with and , or else of the “Type II” form

for some coefficients that are roughly of logarithmic size on the average, and scales with and . As discussed in the previous post, the 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 .

A Type I sum makes a contribution to that can be bounded (via Cauchy-Schwarz) in terms of an expression such as

The inner sum exhibits a lot of cancellation unless is within of an integer. (Here, “a lot” should be loosely interpreted as “gaining many powers of over the trivial bound”.) Since is significantly larger than , standard Vinogradov-type manipulations (see e.g. Lemma 13 of these previous notes) show that this bad case occurs for many only when 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 roughly of the form

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

for ; note that the range is non-trivial because is much larger than . 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 for some . But with , is well below the threshold for the definition of major arc, so we can exclude this case and obtain the required cancellation.

A basic estimate in multiplicative number theory (particularly if one is using the Granville-Soundararajan “pretentious” approach to this subject) is the following inequality of Halasz (formulated here in a quantitative form introduced by Montgomery and Tenenbaum).

Theorem 1 (Halasz inequality)Let be a multiplicative function bounded in magnitude by , and suppose that , , and are such that

As a qualitative corollary, we conclude (by standard compactness arguments) that if

as . In the more recent work of this paper of Granville and Soundararajan, the sharper bound

is obtained (with a more precise description of the term).

The usual proofs of Halasz’s theorem are somewhat lengthy (though there has been a recent simplification, in forthcoming work of Granville, Harper, and Soundarajan). Below the fold I would like to give a relatively short proof of the following “cheap” version of the inequality, which has slightly weaker quantitative bounds, but still suffices to give qualitative conclusions such as (2).

Theorem 2 (Cheap Halasz inequality)Let be a multiplicative function bounded in magnitude by . Let and , and suppose that is sufficiently large depending on . If (1) holds for all , then

The non-optimal exponent can probably be improved a bit by being more careful with the exponents, but I did not try to optimise it here. A similar bound appears in the first paper of Halasz on this topic.

The idea of the argument is to split as a Dirichlet convolution where is the portion of coming from “small”, “medium”, and “large” primes respectively (with the dividing line between the three types of primes being given by various powers of ). Using a Perron-type formula, one can express this convolution in terms of the product of the Dirichlet series of respectively at various complex numbers with . One can use based estimates to control the Dirichlet series of , while using the hypothesis (1) one can get estimates on the Dirichlet series of . (This is similar to the Fourier-analytic approach to ternary additive problems, such as Vinogradov’s theorem on representing large odd numbers as the sum of three primes.) This idea was inspired by a similar device used in the work of Granville, Harper, and Soundarajan. A variant of this argument also appears in unpublished work of Adam Harper.

I thank Andrew Granville for helpful comments which led to significant simplifications of the argument.

Kevin Ford, James Maynard, and I have uploaded to the arXiv our preprint “Chains of large gaps between primes“. This paper was announced in our previous paper with Konyagin and Green, which was concerned with the largest gap

between consecutive primes up to , in which we improved the Rankin bound of

to

for large (where we use the abbreviations , , and ). Here, we obtain an analogous result for the quantity

which measures how far apart the gaps between chains of consecutive primes can be. Our main result is

whenever is sufficiently large depending on , with the implied constant here absolute (and effective). The factor of is inherent to the method, and related to the basic probabilistic fact that if one selects numbers at random from the unit interval , then one expects the minimum gap between adjacent numbers to be about (i.e. smaller than the mean spacing of by an additional factor of ).

Our arguments combine those from the previous paper with the matrix method of Maier, who (in our notation) showed that

for an infinite sequence of going to infinity. (Maier needed to restrict to an infinite sequence to avoid Siegel zeroes, but we are able to resolve this issue by the now standard technique of simply eliminating a prime factor of an exceptional conductor from the sieve-theoretic portion of the argument. As a byproduct, this also makes all of the estimates in our paper effective.)

As its name suggests, the Maier matrix method is usually presented by imagining a matrix of numbers, and using information about the distribution of primes in the columns of this matrix to deduce information about the primes in at least one of the rows of the matrix. We found it convenient to interpret this method in an equivalent probabilistic form as follows. Suppose one wants to find an interval which contained a block of at least primes, each separated from each other by at least (ultimately, will be something like and something like ). One can do this by the probabilistic method: pick to be a random large natural number (with the precise distribution to be chosen later), and try to lower bound the probability that the interval contains at least primes, no two of which are within of each other.

By carefully choosing the residue class of with respect to small primes, one can eliminate several of the from consideration of being prime immediately. For instance, if is chosen to be large and even, then the with even have no chance of being prime and can thus be eliminated; similarly if is large and odd, then cannot be prime for any odd . Using the methods of our previous paper, we can find a residue class (where is a product of a large number of primes) such that, if one chooses to be a large random element of (that is, for some large random integer ), then the set of shifts for which still has a chance of being prime has size comparable to something like ; furthermore this set is fairly well distributed in in the sense that it does not concentrate too strongly in any short subinterval of . The main new difficulty, not present in the previous paper, is to get *lower* bounds on the size of in addition to upper bounds, but this turns out to be achievable by a suitable modification of the arguments.

Using a version of the prime number theorem in arithmetic progressions due to Gallagher, one can show that for each remaining shift , is going to be prime with probability comparable to , so one expects about primes in the set . An upper bound sieve (e.g. the Selberg sieve) also shows that for any distinct , the probability that and are both prime is . Using this and some routine second moment calculations, one can then show that with large probability, the set will indeed contain about primes, no two of which are closer than to each other; with no other numbers in this interval being prime, this gives a lower bound on .

Klaus Roth, who made fundamental contributions to analytic number theory, died this Tuesday, aged 90.

I never met or communicated with Roth personally, but was certainly influenced by his work; he wrote relatively few papers, but they tended to have outsized impact. For instance, he was one of the key people (together with Bombieri) to work on simplifying and generalising the large sieve, taking it from the technically formidable original formulation of Linnik and Rényi to the clean and general almost orthogonality principle that we have today (discussed for instance in these lecture notes of mine). The paper of Roth that had the most impact on my own personal work was his three-page paper proving what is now known as Roth’s theorem on arithmetic progressions:

Theorem 1 (Roth’s theorem on arithmetic progressions)Let be a set of natural numbers of positive upper density (thus ). Then contains infinitely many arithmetic progressions of length three (with non-zero of course).

At the heart of Roth’s elegant argument was the following (surprising at the time) dichotomy: if had some moderately large density within some arithmetic progression , either one could use Fourier-analytic methods to detect the presence of an arithmetic progression of length three inside , or else one could locate a long subprogression of on which had increased density. Iterating this dichotomy by an argument now known as the *density increment argument*, one eventually obtains Roth’s theorem, no matter which side of the dichotomy actually holds. This argument (and the many descendants of it), based on various “dichotomies between structure and randomness”, became essential in many other results of this type, most famously perhaps in Szemerédi’s proof of his celebrated theorem on arithmetic progressions that generalised Roth’s theorem to progressions of arbitrary length. More recently, my recent work on the Chowla and Elliott conjectures that was a crucial component of the solution of the Erdös discrepancy problem, relies on an *entropy decrement argument* which was directly inspired by the density increment argument of Roth.

The Erdös discrepancy problem also is connected with another well known theorem of Roth:

Theorem 2 (Roth’s discrepancy theorem for arithmetic progressions)Let be a sequence in . Then there exists an arithmetic progression in with positive such thatfor an absolute constant .

In fact, Roth proved a stronger estimate regarding mean square discrepancy, which I am not writing down here; as with the Roth theorem in arithmetic progressions, his proof was short and Fourier-analytic in nature (although non-Fourier-analytic proofs have since been found, for instance the semidefinite programming proof of Lovasz). The exponent is known to be sharp (a result of Matousek and Spencer).

As a particular corollary of the above theorem, for an infinite sequence of signs, the sums are unbounded in . The Erdös discrepancy problem asks whether the same statement holds when is restricted to be zero. (Roth also established discrepancy theorems for other sets, such as rectangles, which will not be discussed here.)

Finally, one has to mention Roth’s most famous result, cited for instance in his Fields medal citation:

Theorem 3 (Roth’s theorem on Diophantine approximation)Let be an irrational algebraic number. Then for any there is a quantity such that

From the Dirichlet approximation theorem (or from the theory of continued fractions) we know that the exponent in the denominator cannot be reduced to or below. A classical and easy theorem of Liouville gives the claim with the exponent replaced by the degree of the algebraic number ; work of Thue and Siegel reduced this exponent, but Roth was the one who obtained the near-optimal result. An important point is that the constant is *ineffective* – it is a major open problem in Diophantine approximation to produce any bound significantly stronger than Liouville’s theorem with effective constants. This is because the proof of Roth’s theorem does not exclude any *single* rational from being close to , but instead very ingeniously shows that one cannot have *two* different rationals , that are unusually close to , even when the denominators are very different in size. (I refer to this sort of argument as a “dueling conspiracies” argument; they are strangely prevalent throughout analytic number theory.)

Chantal David, Andrew Granville, Emmanuel Kowalski, Phillipe Michel, Kannan Soundararajan, and I are running a program at MSRI in the Spring of 2017 (more precisely, from Jan 17, 2017 to May 26, 2017) in the area of analytic number theory, with the intention to bringing together many of the leading experts in all aspects of the subject and to present recent work on the many active areas of the subject (e.g. the distribution of the prime numbers, refinements of the circle method, a deeper understanding of the asymptotics of bounded multiplicative functions (and applications to Erdos discrepancy type problems!) and of the “pretentious” approach to analytic number theory, more “analysis-friendly” formulations of the theorems of Deligne and others involving trace functions over fields, and new subconvexity theorems for automorphic forms, to name a few). Like any other semester MSRI program, there will be a number of workshops, seminars, and similar activities taking place while the members are in residence. I’m personally looking forward to the program, which should be occurring in the midst of a particularly productive time for the subject. Needless to say, I (and the rest of the organising committee) plan to be present for most of the program.

Applications for Postdoctoral Fellowships and Research Memberships for this program (and for other MSRI programs in this time period, namely the companion program in Harmonic Analysis and the Fall program in Geometric Group Theory, as well as the complementary program in all other areas of mathematics) remain open until Dec 1. Applications are open to everyone, but require supporting documentation, such as a CV, statement of purpose, and letters of recommendation from other mathematicians; see the application page for more details.

The Chowla conjecture asserts, among other things, that one has the asymptotic

as for any distinct integers , where is the Liouville function. (The usual formulation of the conjecture also allows one to consider more general linear forms than the shifts , but for sake of discussion let us focus on the shift case.) This conjecture remains open for , though there are now some partial results when one averages either in or in the , as discussed in this recent post.

A natural generalisation of the Chowla conjecture is the Elliott conjecture. Its original formulation was basically as follows: one had

whenever were bounded completely multiplicative functions and were distinct integers, and one of the was “non-pretentious” in the sense that

for all Dirichlet characters and real numbers . It is easy to see that some condition like (2) is necessary; for instance if and has period then can be verified to be bounded away from zero as .

In a previous paper with Matomaki and Radziwill, we provided a counterexample to the original formulation of the Elliott conjecture, and proposed that (2) be replaced with the stronger condition

as for any Dirichlet character . To support this conjecture, we proved an averaged and non-asymptotic version of this conjecture which roughly speaking showed a bound of the form

whenever was an arbitrarily slowly growing function of , was sufficiently large (depending on and the rate at which grows), and one of the obeyed the condition

for some that was sufficiently large depending on , and all Dirichlet characters of period at most . As further support of this conjecture, I recently established the bound

under the same hypotheses, where is an arbitrarily slowly growing function of .

In view of these results, it is tempting to conjecture that the condition (4) for one of the should be sufficient to obtain the bound

when is large enough depending on . This may well be the case for . However, the purpose of this blog post is to record a simple counterexample for . Let’s take for simplicity. Let be a quantity much larger than but much smaller than (e.g. ), and set

For , Taylor expansion gives

and

and hence

and hence

On the other hand one can easily verify that all of the obey (4) (the restriction there prevents from getting anywhere close to ). So it seems the correct non-asymptotic version of the Elliott conjecture is the following:

Conjecture 1 (Non-asymptotic Elliott conjecture)Let be a natural number, and let be integers. Let , let be sufficiently large depending on , and let be sufficiently large depending on . Let be bounded multiplicative functions such that for some , one hasfor all Dirichlet characters of conductor at most . Then

The case of this conjecture follows from the work of Halasz; in my recent paper a logarithmically averaged version of the case of this conjecture is established. The requirement to take to be as large as does not emerge in the averaged Elliott conjecture in my previous paper with Matomaki and Radziwill; it thus seems that this averaging has concealed some of the subtler features of the Elliott conjecture. (However, this subtlety does not seem to affect the asymptotic version of the conjecture formulated in that paper, in which the hypothesis is of the form (3), and the conclusion is of the form (1).)

A similar subtlety arises when trying to control the maximal integral

In my previous paper with Matomaki and Radziwill, we could show that easier expression

was small (for a slowly growing function of ) if was bounded and completely multiplicative, and one had a condition of the form

for some large . However, to obtain an analogous bound for (5) it now appears that one needs to strengthen the above condition to

in order to address the counterexample in which for some between and . This seems to suggest that proving (5) (which is closely related to the case of the Chowla conjecture) could in fact be rather difficult; the estimation of (6) relied primarily of prior work of Matomaki and Radziwill which used the hypothesis (7), but as this hypothesis is not sufficient to conclude (5), some additional input must also be used.

Let and be two random variables taking values in the same (discrete) range , and let be some subset of , which we think of as the set of “bad” outcomes for either or . If and have the same probability distribution, then clearly

In particular, if it is rare for to lie in , then it is also rare for to lie in .

If and do not have exactly the same probability distribution, but their probability distributions are *close* to each other in some sense, then we can expect to have an approximate version of the above statement. For instance, from the definition of the total variation distance between two random variables (or more precisely, the total variation distance between the probability distributions of two random variables), we see that

for any . In particular, if it is rare for to lie in , and are close in total variation, then it is also rare for to lie in .

A basic inequality in information theory is Pinsker’s inequality

where the Kullback-Leibler divergence is defined by the formula

(See this previous blog post for a proof of this inequality.) A standard application of Jensen’s inequality reveals that is non-negative (Gibbs’ inequality), and vanishes if and only if , have the same distribution; thus one can think of as a measure of how close the distributions of and are to each other, although one should caution that this is not a symmetric notion of distance, as in general. Inserting Pinsker’s inequality into (1), we see for instance that

Thus, if is close to in the Kullback-Leibler sense, and it is rare for to lie in , then it is rare for to lie in as well.

We can specialise this inequality to the case when a uniform random variable on a finite range of some cardinality , in which case the Kullback-Leibler divergence simplifies to

where

is the Shannon entropy of . Again, a routine application of Jensen’s inequality shows that , with equality if and only if is uniformly distributed on . The above inequality then becomes

Thus, if is a small fraction of (so that it is rare for to lie in ), and the entropy of is very close to the maximum possible value of , then it is rare for to lie in also.

The inequality (2) is only useful when the entropy is close to in the sense that , otherwise the bound is worse than the trivial bound of . In my recent paper on the Chowla and Elliott conjectures, I ended up using a variant of (2) which was still non-trivial when the entropy was allowed to be smaller than . More precisely, I used the following simple inequality, which is implicit in the arguments of that paper but which I would like to make more explicit in this post:

Lemma 1 (Pinsker-type inequality)Let be a random variable taking values in a finite range of cardinality , let be a uniformly distributed random variable in , and let be a subset of . Then

*Proof:* Consider the conditional entropy . On the one hand, we have

by Jensen’s inequality. On the other hand, one has

where we have again used Jensen’s inequality. Putting the two inequalities together, we obtain the claim.

Remark 2As noted in comments, this inequality can be viewed as a special case of the more general inequalityfor arbitrary random variables taking values in the same discrete range , which follows from the data processing inequality

for arbitrary functions , applied to the indicator function . Indeed one has

where is the entropy function.

Thus, for instance, if one has

and

for some much larger than (so that ), then

More informally: if the entropy of is *somewhat* close to the maximum possible value of , and it is *exponentially* rare for a uniform variable to lie in , then it is still *somewhat* rare for to lie in . The estimate given is close to sharp in this regime, as can be seen by calculating the entropy of a random variable which is uniformly distributed inside a small set with some probability and uniformly distributed outside of with probability , for some parameter .

It turns out that the above lemma combines well with concentration of measure estimates; in my paper, I used one of the simplest such estimates, namely Hoeffding’s inequality, but there are of course many other estimates of this type (see e.g. this previous blog post for some others). Roughly speaking, concentration of measure inequalities allow one to make approximations such as

with exponentially high probability, where is a uniform distribution and is some reasonable function of . Combining this with the above lemma, we can then obtain approximations of the form

with somewhat high probability, if the entropy of is somewhat close to maximum. This observation, combined with an “entropy decrement argument” that allowed one to arrive at a situation in which the relevant random variable did have a near-maximum entropy, is the key new idea in my recent paper; for instance, one can use the approximation (3) to obtain an approximation of the form

for “most” choices of and a suitable choice of (with the latter being provided by the entropy decrement argument). The left-hand side is tied to Chowla-type sums such as through the multiplicativity of , while the right-hand side, being a linear correlation involving two parameters rather than just one, has “finite complexity” and can be treated by existing techniques such as the Hardy-Littlewood circle method. One could hope that one could similarly use approximations such as (3) in other problems in analytic number theory or combinatorics.

I’ve just uploaded two related papers to the arXiv:

- The logarithmically averaged Chowla and Elliott conjectures for two-point correlations, submitted to Forum of Mathematics, Pi; and
- The Erdos discrepancy problem, submitted to the new arXiv overlay journal, Discrete Analysis (see this recent announcement on Tim Gowers’ blog).

This pair of papers is an outgrowth of these two recent blog posts and the ensuing discussion. In the first paper, we establish the following logarithmically averaged version of the Chowla conjecture (in the case of two-point correlations (or “pair correlations”)):

Theorem 1 (Logarithmically averaged Chowla conjecture)Let be natural numbers, and let be integers such that . Let be a quantity depending on that goes to infinity as . Let denote the Liouville function. Then one has

For comparison, the non-averaged Chowla conjecture would imply that

which is a strictly stronger estimate than (2), and remains open.

The arguments also extend to other completely multiplicative functions than the Liouville function. In particular, one obtains a slightly averaged version of the non-asymptotic Elliott conjecture that was shown in the previous blog post to imply a positive solution to the Erdos discrepancy problem. The averaged version of the conjecture established in this paper is slightly weaker than the one assumed in the previous blog post, but it turns out that the arguments there can be modified without much difficulty to accept this averaged Elliott conjecture as input. In particular, we obtain an unconditional solution to the Erdos discrepancy problem as a consequence; this is detailed in the second paper listed above. In fact we can also handle the vector-valued version of the Erdos discrepancy problem, in which the sequence takes values in the unit sphere of an arbitrary Hilbert space, rather than in .

Estimates such as (2) or (3) are known to be subject to the “parity problem” (discussed numerous times previously on this blog), which roughly speaking means that they cannot be proven solely using “linear” estimates on functions such as the von Mangoldt function. However, it is known that the parity problem can be circumvented using “bilinear” estimates, and this is basically what is done here.

We now describe in informal terms the proof of Theorem 1, focusing on the model case (2) for simplicity. Suppose for contradiction that the left-hand side of (2) was large and (say) positive. Using the multiplicativity , we conclude that

is also large and positive for all primes that are not too large; note here how the logarithmic averaging allows us to leave the constraint unchanged. Summing in , we conclude that

is large and positive for any given set of medium-sized primes. By a standard averaging argument, this implies that

is large for many choices of , where is a medium-sized parameter at our disposal to choose, and we take to be some set of primes that are somewhat smaller than . (A similar approach was taken in this recent paper of Matomaki, Radziwill, and myself to study sign patterns of the Möbius function.) To obtain the required contradiction, one thus wants to demonstrate significant cancellation in the expression (4). As in that paper, we view as a random variable, in which case (4) is essentially a bilinear sum of the random sequence along a random graph on , in which two vertices are connected if they differ by a prime in that divides . A key difficulty in controlling this sum is that for randomly chosen , the sequence and the graph need not be independent. To get around this obstacle we introduce a new argument which we call the “entropy decrement argument” (in analogy with the “density increment argument” and “energy increment argument” that appear in the literature surrounding Szemerédi’s theorem on arithmetic progressions, and also reminiscent of the “entropy compression argument” of Moser and Tardos, discussed in this previous post). This argument, which is a simple consequence of the Shannon entropy inequalities, can be viewed as a quantitative version of the standard subadditivity argument that establishes the existence of Kolmogorov-Sinai entropy in topological dynamical systems; it allows one to select a scale parameter (in some suitable range ) for which the sequence and the graph exhibit some weak independence properties (or more precisely, the mutual information between the two random variables is small).

Informally, the entropy decrement argument goes like this: if the sequence has significant mutual information with , then the entropy of the sequence for will grow a little slower than linearly, due to the fact that the graph has zero entropy (knowledge of more or less completely determines the shifts of the graph); this can be formalised using the classical Shannon inequalities for entropy (and specifically, the non-negativity of conditional mutual information). But the entropy cannot drop below zero, so by increasing as necessary, at some point one must reach a metastable region (cf. the finite convergence principle discussed in this previous blog post), within which very little mutual information can be shared between the sequence and the graph . Curiously, for the application it is not enough to have a purely quantitative version of this argument; one needs a quantitative bound (which gains a factor of a bit more than on the trivial bound for mutual information), and this is surprisingly delicate (it ultimately comes down to the fact that the series diverges, which is only barely true).

Once one locates a scale with the low mutual information property, one can use standard concentration of measure results such as the Hoeffding inequality to approximate (4) by the significantly simpler expression

The important thing here is that Hoeffding’s inequality gives exponentially strong bounds on the failure probability, which is needed to counteract the logarithms that are inevitably present whenever trying to use entropy inequalities. The expression (5) can then be controlled in turn by an application of the Hardy-Littlewood circle method and a non-trivial estimate

for averaged short sums of a modulated Liouville function established in another recent paper by Matomäki, Radziwill and myself.

When one uses this method to study more general sums such as

one ends up having to consider expressions such as

where is the coefficient . When attacking this sum with the circle method, one soon finds oneself in the situation of wanting to locate the large Fourier coefficients of the exponential sum

In many cases (such as in the application to the Erdös discrepancy problem), the coefficient is identically , and one can understand this sum satisfactorily using the classical results of Vinogradov: basically, is large when lies in a “major arc” and is small when it lies in a “minor arc”. For more general functions , the coefficients are more or less arbitrary; the large values of are no longer confined to the major arc case. Fortunately, even in this general situation one can use a restriction theorem for the primes established some time ago by Ben Green and myself to show that there are still only a bounded number of possible locations (up to the uncertainty mandated by the Heisenberg uncertainty principle) where is large, and we can still conclude by using (6). (Actually, as recently pointed out to me by Ben, one does not need the full strength of our result; one only needs the restriction theorem for the primes, which can be proven fairly directly using Plancherel’s theorem and some sieve theory.)

It is tempting to also use the method to attack higher order cases of the (logarithmically) averaged Chowla conjecture, for instance one could try to prove the estimate

The above arguments reduce matters to obtaining some non-trivial cancellation for sums of the form

A little bit of “higher order Fourier analysis” (as was done for very similar sums in the ergodic theory context by Frantzikinakis-Host-Kra and Wooley-Ziegler) lets one control this sort of sum if one can establish a bound of the form

where goes to infinity and is a very slowly growing function of . This looks very similar to (6), but the fact that the supremum is now inside the integral makes the problem much more difficult. However it looks worth attacking (7) further, as this estimate looks like it should have many nice applications (beyond just the case of the logarithmically averaged Chowla or Elliott conjectures, which is already interesting).

For higher than , the same line of analysis requires one to replace the linear phase by more complicated phases, such as quadratic phases or even -step nilsequences. Given that (7) is already beyond the reach of current literature, these even more complicated expressions are also unavailable at present, but one can imagine that they will eventually become tractable, in which case we would obtain an averaged form of the Chowla conjecture for all , which would have a number of consequences (such as a logarithmically averaged version of Sarnak’s conjecture, as per this blog post).

It would of course be very nice to remove the logarithmic averaging, and be able to establish bounds such as (3). I did attempt to do so, but I do not see a way to use the entropy decrement argument in a manner that does not require some sort of averaging of logarithmic type, as it requires one to pick a scale that one cannot specify in advance, which is not a problem for logarithmic averages (which are quite stable with respect to dilations) but is problematic for ordinary averages. But perhaps the problem can be circumvented by some clever modification of the argument. One possible approach would be to start exploiting multiplicativity at products of primes, and not just individual primes, to try to keep the scale fixed, but this makes the concentration of measure part of the argument much more complicated as one loses some independence properties (coming from the Chinese remainder theorem) which allowed one to conclude just from the Hoeffding inequality.

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

as for any fixed natural number . 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

For such correlations, the conjecture was that one had

as for any natural number , as long as was a completely multiplicative function with magnitude bounded by , and such that

for any Dirichlet character and any real number . In the language of “pretentious number theory”, as developed by Granville and Soundararajan, the hypothesis (2) asserts that the completely multiplicative function does not “pretend” to be like the completely multiplicative function for any character and real number . A condition of this form is necessary; for instance, if is precisely equal to and has period , then is equal to as 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 having the property that “pretends” *locally* to be the function for in various intervals , where and 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 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

as for all fixed Dirichlet characters . 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 parameter does not go to infinity, or equivalently that the function is permitted to depend on :

Conjecture 1 (Non-asymptotic Elliott conjecture)Let , let be sufficiently large depending on , and let be sufficiently large depending on . Suppose that is a completely multiplicative function with magnitude bounded by , such thatfor all Dirichlet characters of period at most . Then one has

for all natural numbers .

The -dependent factor in the constraint is necessary, as can be seen by considering the completely multiplicative function (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 , the discrepancyis 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 that takes values in rather than . For this function, one has from the complete multiplicativity of that

If denotes the period of , then has mean zero on every interval of length , and thus

Thus has bounded discrepancy.

Of course, this is not a true counterexample to Conjecture 2 because can take the value . 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 be the non-principal Dirichlet character of period (thus equals when , when , and when ), and define the completely multiplicative function by setting when and . This is about the simplest modification one can make to the previous near-counterexample to eliminate the zeroes. Now consider the sum

with for some large . Writing with coprime to and at most , we can write this sum as

Now observe that . The function has mean zero on every interval of length three, and is equal to mod , and thus

for every , and thus

Thus also has unbounded discrepancy, but only barely so (it grows logarithmically in ). These examples suggest that the main “enemy” to proving Conjecture 2 comes from completely multiplicative functions 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 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 . 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 must pretend to be like a function of the form . 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.)

The twin prime conjecture is one of the oldest unsolved problems in analytic number theory. There are several reasons why this conjecture remains out of reach of current techniques, but the most important obstacle is the parity problem which prevents purely sieve-theoretic methods (or many other popular methods in analytic number theory, such as the circle method) from detecting pairs of prime twins in a way that can distinguish them from other twins of almost primes. The parity problem is discussed in these previous blog posts; this obstruction is ultimately powered by the *Möbius pseudorandomness principle* that asserts that the Möbius function is asymptotically orthogonal to all “structured” functions (and in particular, to the weight functions constructed from sieve theory methods).

However, there is an intriguing “alternate universe” in which the Möbius function *is* strongly correlated with some structured functions, and specifically with some Dirichlet characters, leading to the existence of the infamous “Siegel zero“. In this scenario, the parity problem obstruction disappears, and it becomes possible, *in principle*, to attack problems such as the twin prime conjecture. In particular, we have the following result of Heath-Brown:

Theorem 1At least one of the following two statements are true:

- (Twin prime conjecture) There are infinitely many primes such that is also prime.
- (No Siegel zeroes) There exists a constant such that for every real Dirichlet character of conductor , the associated Dirichlet -function has no zeroes in the interval .

Informally, this result asserts that if one had an infinite sequence of Siegel zeroes, one could use this to generate infinitely many twin primes. See this survey of Friedlander and Iwaniec for more on this “illusory” or “ghostly” parallel universe in analytic number theory that should not actually exist, but is surprisingly self-consistent and to date proven to be impossible to banish from the realm of possibility.

The strategy of Heath-Brown’s proof is fairly straightforward to describe. The usual starting point is to try to lower bound

for some large value of , where is the von Mangoldt function. Actually, in this post we will work with the slight variant

where

is the second von Mangoldt function, and denotes Dirichlet convolution, and is an (unsquared) Selberg sieve that damps out small prime factors. This sum also detects twin primes, but will lead to slightly simpler computations. For technical reasons we will also smooth out the interval and remove very small primes from , but we will skip over these steps for the purpose of this informal discussion. (In Heath-Brown’s original paper, the Selberg sieve is essentially replaced by the more combinatorial restriction for some large , where is the primorial of , but I found the computations to be slightly easier if one works with a Selberg sieve, particularly if the sieve is not squared to make it nonnegative.)

If there is a Siegel zero with close to and a Dirichlet character of conductor , then multiplicative number theory methods can be used to show that the Möbius function “pretends” to be like the character in the sense that for “most” primes near (e.g. in the range for some small and large ). Traditionally, one uses complex-analytic methods to demonstrate this, but one can also use elementary multiplicative number theory methods to establish these results (qualitatively at least), as will be shown below the fold.

The fact that pretends to be like can be used to construct a tractable approximation (after inserting the sieve weight ) in the range (where for some large ) for the second von Mangoldt function , namely the function

Roughly speaking, we think of the periodic function and the slowly varying function as being of about the same “complexity” as the constant function , so that is roughly of the same “complexity” as the divisor function

which is considerably simpler to obtain asymptotics for than the von Mangoldt function as the Möbius function is no longer present. (For instance, note from the Dirichlet hyperbola method that one can estimate to accuracy with little difficulty, whereas to obtain a comparable level of accuracy for or is essentially the Riemann hypothesis.)

One expects to be a good approximant to if is of size and has no prime factors less than for some large constant . The Selberg sieve will be mostly supported on numbers with no prime factor less than . As such, one can hope to approximate (1) by the expression

as it turns out, the error between this expression and (1) is easily controlled by sieve-theoretic techniques. Let us ignore the Selberg sieve for now and focus on the slightly simpler sum

As discussed above, this sum should be thought of as a slightly more complicated version of the sum

Accordingly, let us look (somewhat informally) at the task of estimating the model sum (3). One can think of this problem as basically that of counting solutions to the equation with in various ranges; this is clearly related to understanding the equidistribution of the hyperbola in . Taking Fourier transforms, the latter problem is closely related to estimation of the Kloosterman sums

where denotes the inverse of in . One can then use the Weil bound

where is the greatest common divisor of (with the convention that this is equal to if vanish), and the decays to zero as . The Weil bound yields good enough control on error terms to estimate (3), and as it turns out the same method also works to estimate (2) (provided that with large enough).

Actually one does not need the full strength of the Weil bound here; any power savings over the trivial bound of will do. In particular, it will suffice to use the weaker, but easier to prove, bounds of Kloosterman:

Lemma 2 (Kloosterman bound)One has

whenever and are coprime to , where the is with respect to the limit (and is uniform in ).

*Proof:* Observe from change of variables that the Kloosterman sum is unchanged if one replaces with for . For fixed , the number of such pairs is at least , thanks to the divisor bound. Thus it will suffice to establish the fourth moment bound

The left-hand side can be rearranged as

which by Fourier summation is equal to

Observe from the quadratic formula and the divisor bound that each pair has at most solutions to the system of equations . Hence the number of quadruples of the desired form is , and the claim follows.

We will also need another easy case of the Weil bound to handle some other portions of (2):

Lemma 3 (Easy Weil bound)Let be a primitive real Dirichlet character of conductor , and let . Then

*Proof:* As is the conductor of a primitive real Dirichlet character, is equal to times a squarefree odd number for some . By the Chinese remainder theorem, it thus suffices to establish the claim when is an odd prime. We may assume that is not divisible by this prime , as the claim is trivial otherwise. If vanishes then does not vanish, and the claim follows from the mean zero nature of ; similarly if vanishes. Hence we may assume that do not vanish, and then we can normalise them to equal . By completing the square it now suffices to show that

whenever . As is on the quadratic residues and on the non-residues, it now suffices to show that

But by making the change of variables , the left-hand side becomes , and the claim follows.

While the basic strategy of Heath-Brown’s argument is relatively straightforward, implementing it requires a large amount of computation to control both main terms and error terms. I experimented for a while with rearranging the argument to try to reduce the amount of computation; I did not fully succeed in arriving at a satisfactorily minimal amount of superfluous calculation, but I was able to at least reduce this amount a bit, mostly by replacing a combinatorial sieve with a Selberg-type sieve (which was not needed to be positive, so I dispensed with the squaring aspect of the Selberg sieve to simplify the calculations a little further; also for minor reasons it was convenient to retain a tiny portion of the combinatorial sieve to eliminate extremely small primes). Also some modest reductions in complexity can be obtained by using the second von Mangoldt function in place of . These exercises were primarily for my own benefit, but I am placing them here in case they are of interest to some other readers.

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