Value patterns of multiplicative functions and related sequences

Joni Teräväinen and I have just uploaded to the arXiv our paper “Value patterns of multiplicative functions and related sequences“, submitted to Forum of Mathematics, Sigma. This paper explores how to use recent technology on correlations of multiplicative (or nearly multiplicative functions), such as the “entropy decrement method”, in conjunction with techniques from additive combinatorics, to establish new results on the sign patterns of functions such as the Liouville function . For instance, with regards to length 5 sign patterns

of the Liouville function, we can now show that at least of the possible sign patterns in occur with positive upper density. (Conjecturally, all of them do so, and this is known for all shorter sign patterns, but unfortunately seems to be the limitation of our methods.)

The Liouville function can be written as , where is the number of prime factors of (counting multiplicity). One can also consider the variant , which is a completely multiplicative function taking values in the cube roots of unity . Here we are able to show that all sign patterns in occur with positive lower density as sign patterns of this function. The analogous result for was already known (see this paper of Matomäki, Radziwiłł, and myself), and in that case it is even known that all sign patterns occur with equal logarithmic density (from this paper of myself and Teräväinen), but these techniques barely fail to handle the case by itself (largely because the “parity” arguments used in the case of the Liouville function no longer control three-point correlations in the case) and an additional additive combinatorial tool is needed. After applying existing technology (such as entropy decrement methods), the problem roughly speaking reduces to locating patterns for a certain partition of a compact abelian group (think for instance of the unit circle , although the general case is a bit more complicated, in particular if is disconnected then there is a certain “coprimality” constraint on , also we can allow the to be replaced by any with divisible by ), with each of the having measure . An inequality of Kneser just barely fails to guarantee the existence of such patterns, but by using an inverse theorem for Kneser’s inequality in this previous paper of mine we are able to identify precisely the obstruction for this method to work, and rule it out by an ad hoc method.

The same techniques turn out to also make progress on some conjectures of Erdös-Pomerance and Hildebrand regarding patterns of the largest prime factor of a natural number . For instance, we improve results of Erdös-Pomerance and of Balog demonstrating that the inequalities

and

each hold for infinitely many , by demonstrating the stronger claims that the inequalities

and

each hold for a set of of positive lower density. As a variant, we also show that we can find a positive density set of for which

for any fixed (this improves on a previous result of Hildebrand with replaced by . A number of other results of this type are also obtained in this paper.

In order to obtain these sorts of results, one needs to extend the entropy decrement technology from the setting of multiplicative functions to that of what we call “weakly stable sets” – sets which have some multiplicative structure, in the sense that (roughly speaking) there is a set such that for all small primes , the statements and are roughly equivalent to each other. For instance, if is a level set , one would take ; if instead is a set of the form , then one can take . When one has such a situation, then very roughly speaking, the entropy decrement argument then allows one to estimate a one-parameter correlation such as

with a two-parameter correlation such as

(where we will be deliberately vague as to how we are averaging over and ), and then the use of the “linear equations in primes” technology of Ben Green, Tamar Ziegler, and myself then allows one to replace this average in turn by something like

where is constrained to be not divisible by small primes but is otherwise quite arbitrary. This latter average can then be attacked by tools from additive combinatorics, such as translation to a continuous group model (using for instance the Furstenberg correspondence principle) followed by tools such as Kneser’s inequality (or inverse theorems to that inequality).

The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures

Joni Teräväinen and I have just uploaded to the arXiv our paper “The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures“. This is a sequel to our previous paper that studied logarithmic correlations of the form

where were bounded multiplicative functions, were fixed shifts, was a quantity going off to infinity, and was a generalised limit functional. Our main technical result asserted that these correlations were necessarily the uniform limit of periodic functions . Furthermore, if (weakly) pretended to be a Dirichlet character , then the could be chosen to be –isotypic in the sense that whenever are integers with coprime to the periods of and ; otherwise, if did not weakly pretend to be any Dirichlet character , then vanished completely. This was then used to verify several cases of the logarithmically averaged Elliott and Chowla conjectures.

The purpose of this paper was to investigate the extent to which the methods could be extended to non-logarithmically averaged settings. For our main technical result, we now considered the unweighted averages

where is an additional parameter. Our main result was now as follows. If did not weakly pretend to be a twisted Dirichlet character , then converged to zero on (doubly logarithmic) average as . If instead did pretend to be such a twisted Dirichlet character, then converged on (doubly logarithmic) average to a limit of -isotypic functions . Thus, roughly speaking, one has the approximation

for most .

Informally, this says that at almost all scales (where “almost all” means “outside of a set of logarithmic density zero”), the non-logarithmic averages behave much like their logarithmic counterparts except for a possible additional twisting by an Archimedean character (which interacts with the Archimedean parameter in much the same way that the Dirichlet character interacts with the non-Archimedean parameter ). One consequence of this is that most of the recent results on the logarithmically averaged Chowla and Elliott conjectures can now be extended to their non-logarithmically averaged counterparts, so long as one excludes a set of exceptional scales of logarithmic density zero. For instance, the Chowla conjecture

is now established for either odd or equal to , so long as one excludes an exceptional set of scales.

In the logarithmically averaged setup, the main idea was to combine two very different pieces of information on . The first, coming from recent results in ergodic theory, was to show that was well approximated in some sense by a nilsequence. The second was to use the “entropy decrement argument” to obtain an approximate isotopy property of the form

for “most” primes and integers . Combining the two facts, one eventually finds that only the almost periodic components of the nilsequence are relevant.

In the current situation, each is approximated by a nilsequence, but the nilsequence can vary with (although there is some useful “Lipschitz continuity” of this nilsequence with respect to the parameter). Meanwhile, the entropy decrement argument gives an approximation basically of the form

for “most” . The arguments then proceed largely as in the logarithmically averaged case. A key lemma to handle the dependence on the new parameter is the following cohomological statement: if one has a map that was a quasimorphism in the sense that for all and some small , then there exists a real number such that for all small . This is achieved by applying a standard “cocycle averaging argument” to the cocycle .

It would of course be desirable to not have the set of exceptional scales. We only know of one (implausible) scenario in which we can do this, namely when one has far fewer (in particular, subexponentially many) sign patterns for (say) the Liouville function than predicted by the Chowla conjecture. In this scenario (roughly analogous to the “Siegel zero” scenario in multiplicative number theory), the entropy of the Liouville sign patterns is so small that the entropy decrement argument becomes powerful enough to control all scales rather than almost all scales. On the other hand, this scenario seems to be self-defeating, in that it allows one to establish a large number of cases of the Chowla conjecture, and the full Chowla conjecture is inconsistent with having unusually few sign patterns. Still it hints that future work in this direction may need to split into “low entropy” and “high entropy” cases, in analogy to how many arguments in multiplicative number theory have to split into the “Siegel zero” and “no Siegel zero” cases.

Odd order cases of the logarithmically averaged Chowla conjecture

Joni Teräväinen and I have just uploaded to the arXiv our paper “Odd order cases of the logarithmically averaged Chowla conjecture“, submitted to J. Numb. Thy. Bordeaux. This paper gives an alternate route to one of the main results of our previous paper, and more specifically reproves the asymptotic

 

for all odd and all integers (that is to say, all the odd order cases of the logarithmically averaged Chowla conjecture). Our previous argument relies heavily on some deep ergodic theory results of Bergelson-Host-Kra, Leibman, and Le (and was applicable to more general multiplicative functions than the Liouville function ); here we give a shorter proof that avoids ergodic theory (but instead requires the Gowers uniformity of the (W-tricked) von Mangoldt function, established in several papers of Ben Green, Tamar Ziegler, and myself). The proof follows the lines sketched in the previous blog post. In principle, due to the avoidance of ergodic theory, the arguments here have a greater chance to be made quantitative; however, at present the known bounds on the Gowers uniformity of the von Mangoldt function are qualitative, except at the level, which is unfortunate since the first non-trivial odd case requires quantitative control on the level. (But it may be possible to make the Gowers uniformity bounds for quantitative if one assumes GRH, although when one puts everything together, the actual decay rate obtained in (1) is likely to be poor.)

The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures

Joni Teräväinen and I have just uploaded to the arXiv our paper “The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures“, submitted to Duke Mathematical Journal. This paper builds upon my previous paper in which I introduced an “entropy decrement method” to prove the two-point (logarithmically averaged) cases of the Chowla and Elliott conjectures. A bit more specifically, I showed that

whenever were sequences going to infinity, were distinct integers, and were -bounded multiplicative functions which were non-pretentious in the sense that

for all Dirichlet characters and for . Thus, for instance, one had the logarithmically averaged two-point Chowla conjecture

for fixed any non-zero , where was the Liouville function.

One would certainly like to extend these results to higher order correlations than the two-point correlations. This looks to be difficult (though perhaps not completely impossible if one allows for logarithmic averaging): in a previous paper I showed that achieving this in the context of the Liouville function would be equivalent to resolving the logarithmically averaged Sarnak conjecture, as well as establishing logarithmically averaged local Gowers uniformity of the Liouville function. However, in this paper we are able to avoid having to resolve these difficult conjectures to obtain partial results towards the (logarithmically averaged) Chowla and Elliott conjecture. For the Chowla conjecture, we can obtain all odd order correlations, in that

for all odd and all integers (which, in the odd order case, are no longer required to be distinct). (Superficially, this looks like we have resolved “half of the cases” of the logarithmically averaged Chowla conjecture; but it seems the odd order correlations are significantly easier than the even order ones. For instance, because of the Katai-Bourgain-Sarnak-Ziegler criterion, one can basically deduce the odd order cases of (2) from the even order cases (after allowing for some dilations in the argument ).

For the more general Elliott conjecture, we can show that

for any , any integers and any bounded multiplicative functions , unless the product weakly pretends to be a Dirichlet character in the sense that

This can be seen to imply (2) as a special case. Even when does pretend to be a Dirichlet character , we can still say something: if the limits

exist for each (which can be guaranteed if we pass to a suitable subsequence), then is the uniform limit of periodic functions , each of which is –isotypic in the sense that whenever are integers with coprime to the periods of and . This does not pin down the value of any single correlation , but does put significant constraints on how these correlations may vary with .

Among other things, this allows us to show that all possible length four sign patterns of the Liouville function occur with positive density, and all possible length four sign patterns occur with the conjectured logarithmic density. (In a previous paper with Matomaki and Radziwill, we obtained comparable results for length three patterns of Liouville and length two patterns of Möbius.)

To describe the argument, let us focus for simplicity on the case of the Liouville correlations

assuming for sake of discussion that all limits exist. (In the paper, we instead use the device of generalised limits, as discussed in this previous post.) The idea is to combine together two rather different ways to control this function . The first proceeds by the entropy decrement method mentioned earlier, which roughly speaking works as follows. Firstly, we pick a prime and observe that for any , which allows us to rewrite (3) as

Making the change of variables , we obtain

The difference between and is negligible in the limit (here is where we crucially rely on the log-averaging), hence

and thus by (3) we have

The entropy decrement argument can be used to show that the latter limit is small for most (roughly speaking, this is because the factors behave like independent random variables as varies, so that concentration of measure results such as Hoeffding’s inequality can apply, after using entropy inequalities to decouple somewhat these random variables from the factors). We thus obtain the approximate isotopy property

for most and .

On the other hand, by the Furstenberg correspondence principle (as discussed in these previous posts), it is possible to express as a multiple correlation

for some probability space equipped with a measure-preserving invertible map . Using results of Bergelson-Host-Kra, Leibman, and Le, this allows us to obtain a decomposition of the form

where is a nilsequence, and goes to zero in density (even along the primes, or constant multiples of the primes). The original work of Bergelson-Host-Kra required ergodicity on , which is very definitely a hypothesis that is not available here; however, the later work of Leibman removed this hypothesis, and the work of Le refined the control on so that one still has good control when restricting to primes, or constant multiples of primes.

Ignoring the small error , we can now combine (5) to conclude that

Using the equidistribution theory of nilsequences (as developed in this previous paper of Ben Green and myself), one can break up further into a periodic piece and an “irrational” or “minor arc” piece . The contribution of the minor arc piece can be shown to mostly cancel itself out after dilating by primes and averaging, thanks to Vinogradov-type bilinear sum estimates (transferred to the primes). So we end up with

which already shows (heuristically, at least) the claim that can be approximated by periodic functions which are isotopic in the sense that

But if is odd, one can use Dirichlet’s theorem on primes in arithmetic progressions to restrict to primes that are modulo the period of , and conclude now that vanishes identically, which (heuristically, at least) gives (2).

The same sort of argument works to give the more general bounds on correlations of bounded multiplicative functions. But for the specific task of proving (2), we initially used a slightly different argument that avoids using the ergodic theory machinery of Bergelson-Host-Kra, Leibman, and Le, but replaces it instead with the Gowers uniformity norm theory used to count linear equations in primes. Basically, by averaging (4) in using the “-trick”, as well as known facts about the Gowers uniformity of the von Mangoldt function, one can obtain an approximation of the form

where ranges over a large range of integers coprime to some primorial . On the other hand, by iterating (4) we have

for most semiprimes , and by again averaging over semiprimes one can obtain an approximation of the form

For odd, one can combine the two approximations to conclude that . (This argument is not given in the current paper, but we plan to detail it in a subsequent one.)