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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

\displaystyle  f(a) := \lim^*_{x \rightarrow \infty} \frac{1}{\log \omega(x)} \sum_{x/\omega(x) \leq n \leq x} \frac{g_1(n+ah_1) \dots g_k(n+ah_k)}{n},

where {g_1,\dots,g_k} were bounded multiplicative functions, {h_1,\dots,h_k \rightarrow \infty} were fixed shifts, {1 \leq \omega(x) \leq x} was a quantity going off to infinity, and {\lim^*} was a generalised limit functional. Our main technical result asserted that these correlations were necessarily the uniform limit of periodic functions {f_i}. Furthermore, if {g_1 \dots g_k} (weakly) pretended to be a Dirichlet character {\chi}, then the {f_i} could be chosen to be {\chi}isotypic in the sense that {f_i(ab) = f_i(a) \chi(b)} whenever {a,b} are integers with {b} coprime to the periods of {\chi} and {f_i}; otherwise, if {g_1 \dots g_k} did not weakly pretend to be any Dirichlet character {\chi}, then {f} 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

\displaystyle  f_d(a) := \lim^*_{x \rightarrow \infty} \frac{1}{x/d} \sum_{n \leq x/d} g_1(n+ah_1) \dots g_k(n+ah_k),

where {d>1} is an additional parameter. Our main result was now as follows. If {g_1 \dots g_k} did not weakly pretend to be a twisted Dirichlet character {n \mapsto \chi(n) n^{it}}, then {f_d(a)} converged to zero on (doubly logarithmic) average as {d \rightarrow \infty}. If instead {g_1 \dots g_k} did pretend to be such a twisted Dirichlet character, then {f_d(a) d^{it}} converged on (doubly logarithmic) average to a limit {f(a)} of {\chi}-isotypic functions {f_i}. Thus, roughly speaking, one has the approximation

\displaystyle  \lim^*_{x \rightarrow \infty} \frac{1}{x/d} \sum_{n \leq x/d} g_1(n+ah_1) \dots g_k(n+ah_k) \approx f(a) d^{-it}

for most {d}.

Informally, this says that at almost all scales {x} (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 {d \mapsto d^{it}} (which interacts with the Archimedean parameter {d} in much the same way that the Dirichlet character {\chi} interacts with the non-Archimedean parameter {a}). 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 {x} of logarithmic density zero. For instance, the Chowla conjecture

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

is now established for {k} either odd or equal to {2}, 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 {f(a)}. The first, coming from recent results in ergodic theory, was to show that {f(a)} 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

\displaystyle  f(a) g_1 \dots g_k(p)\approx f(ap)

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

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

\displaystyle  f_{dp}(a) g_1 \dots g_k(p)\approx f_d(ap)

for “most” {d,p,a}. The arguments then proceed largely as in the logarithmically averaged case. A key lemma to handle the dependence on the new parameter {d} is the following cohomological statement: if one has a map {\alpha: (0,+\infty) \rightarrow S^1} that was a quasimorphism in the sense that {\alpha(xy) = \alpha(x) \alpha(y) + O(\varepsilon)} for all {x,y \in (0,+\infty)} and some small {\varepsilon}, then there exists a real number {t} such that {\alpha(x) = x^{it} + O(\varepsilon)} for all small {\varepsilon}. This is achieved by applying a standard “cocycle averaging argument” to the cocycle {(x,y) \mapsto \alpha(xy) \alpha(x)^{-1} \alpha(y)^{-1}}.

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

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

\displaystyle \sum_{n \leq x} \frac{\lambda(n+h_1) \dots \lambda(n+h_k)}{n} = o(\log x) \ \ \ \ \ (1)

 

for all odd {k} and all integers {h_1,\dots,h_k} (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 {\lambda}); 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 {U^2} level, which is unfortunate since the first non-trivial odd case {k=3} requires quantitative control on the {U^3} level. (But it may be possible to make the Gowers uniformity bounds for {U^3} quantitative if one assumes GRH, although when one puts everything together, the actual decay rate obtained in (1) is likely to be poor.)

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

\displaystyle  \lim_{m \rightarrow \infty} \frac{1}{\log \omega_m} \sum_{x_m/\omega_m \leq n \leq x_m} \frac{g_0(n+h_0) g_1(n+h_1)}{n} = 0

whenever {1 \leq \omega_m \leq x_m} were sequences going to infinity, {h_0,h_1} were distinct integers, and {g_0,g_1: {\bf N} \rightarrow {\bf C}} were {1}-bounded multiplicative functions which were non-pretentious in the sense that

\displaystyle  \liminf_{X \rightarrow \infty} \inf_{|t_j| \leq X} \sum_{p \leq X} \frac{1-\mathrm{Re}( g_j(p) \overline{\chi_j}(p) p^{it_j})}{p} = \infty \ \ \ \ \ (1)

for all Dirichlet characters {\chi_j} and for {j=0,1}. Thus, for instance, one had the logarithmically averaged two-point Chowla conjecture

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

for fixed any non-zero {h}, where {\lambda} 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

\displaystyle  \sum_{n \leq x} \frac{\lambda(n+h_1) \dots \lambda(n+h_k)}{n} = o(\log x) \ \ \ \ \ (2)

for all odd {k} and all integers {h_1,\dots,h_k} (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 {n}).

For the more general Elliott conjecture, we can show that

\displaystyle  \lim_{m \rightarrow \infty} \frac{1}{\log \omega_m} \sum_{x_m/\omega_m \leq n \leq x_m} \frac{g_1(n+h_1) \dots g_k(n+h_k)}{n} = 0

for any {k}, any integers {h_1,\dots,h_k} and any bounded multiplicative functions {g_1,\dots,g_k}, unless the product {g_1 \dots g_k} weakly pretends to be a Dirichlet character {\chi} in the sense that

\displaystyle  \sum_{p \leq X} \frac{1 - \hbox{Re}( g_1 \dots g_k(p) \overline{\chi}(p)}{p} = o(\log\log X).

This can be seen to imply (2) as a special case. Even when {g_1,\dots,g_k} does pretend to be a Dirichlet character {\chi}, we can still say something: if the limits

\displaystyle  f(a) := \lim_{m \rightarrow \infty} \frac{1}{\log \omega_m} \sum_{x_m/\omega_m \leq n \leq x_m} \frac{g_1(n+ah_1) \dots g_k(n+ah_k)}{n}

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

Among other things, this allows us to show that all {16} possible length four sign patterns {(\lambda(n+1),\dots,\lambda(n+4)) \in \{-1,+1\}^4} of the Liouville function occur with positive density, and all {65} possible length four sign patterns {(\mu(n+1),\dots,\mu(n+4)) \in \{-1,0,+1\}^4 \backslash \{-1,+1\}^4} 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

\displaystyle  f(a) := \lim_{X \rightarrow \infty} \frac{1}{\log X} \sum_{n \leq X} \frac{\lambda(n) \lambda(n+a) \dots \lambda(n+(k-1)a)}{n}, \ \ \ \ \ (3)

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 {f}. The first proceeds by the entropy decrement method mentioned earlier, which roughly speaking works as follows. Firstly, we pick a prime {p} and observe that {\lambda(pn)=-\lambda(n)} for any {n}, which allows us to rewrite (3) as

\displaystyle  (-1)^k f(a) = \lim_{X \rightarrow \infty} \frac{1}{\log X}

\displaystyle  \sum_{n \leq X} \frac{\lambda(pn) \lambda(pn+ap) \dots \lambda(pn+(k-1)ap)}{n}.

Making the change of variables {n' = pn}, we obtain

\displaystyle  (-1)^k f(a) = \lim_{X \rightarrow \infty} \frac{1}{\log X}

\displaystyle \sum_{n' \leq pX} \frac{\lambda(n') \lambda(n'+ap) \dots \lambda(n'+(k-1)ap)}{n'} p 1_{p|n'}.

The difference between {n' \leq pX} and {n' \leq X} is negligible in the limit (here is where we crucially rely on the log-averaging), hence

\displaystyle  (-1)^k f(a) = \lim_{X \rightarrow \infty} \frac{1}{\log X} \sum_{n \leq X} \frac{\lambda(n) \lambda(n+ap) \dots \lambda(n+(k-1)ap)}{n} p 1_{p|n}

and thus by (3) we have

\displaystyle  (-1)^k f(a) = f(ap) + \lim_{X \rightarrow \infty} \frac{1}{\log X}

\displaystyle \sum_{n \leq X} \frac{\lambda(n) \lambda(n+ap) \dots \lambda(n+(k-1)ap)}{n} (p 1_{p|n}-1).

The entropy decrement argument can be used to show that the latter limit is small for most {p} (roughly speaking, this is because the factors {p 1_{p|n}-1} behave like independent random variables as {p} 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 {\lambda} factors). We thus obtain the approximate isotopy property

\displaystyle  (-1)^k f(a) \approx f(ap) \ \ \ \ \ (4)

for most {a} and {p}.

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

\displaystyle  f(a) = \int_X g(x) g(T^a x) \dots g(T^{(k-1)a} x)\ d\mu(x)

for some probability space {(X,\mu)} equipped with a measure-preserving invertible map {T: X \rightarrow X}. Using results of Bergelson-Host-Kra, Leibman, and Le, this allows us to obtain a decomposition of the form

\displaystyle  f(a) = f_1(a) + f_2(a) \ \ \ \ \ (5)

where {f_1} is a nilsequence, and {f_2} 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 {X}, 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 {f_1} so that one still has good control when restricting to primes, or constant multiples of primes.

Ignoring the small error {f_2(a)}, we can now combine (5) to conclude that

\displaystyle  f(a) \approx (-1)^k f_1(ap).

Using the equidistribution theory of nilsequences (as developed in this previous paper of Ben Green and myself), one can break up {f_1} further into a periodic piece {f_0} and an “irrational” or “minor arc” piece {f_3}. The contribution of the minor arc piece {f_3} can be shown to mostly cancel itself out after dilating by primes {p} and averaging, thanks to Vinogradov-type bilinear sum estimates (transferred to the primes). So we end up with

\displaystyle  f(a) \approx (-1)^k f_0(ap),

which already shows (heuristically, at least) the claim that {f} can be approximated by periodic functions {f_0} which are isotopic in the sense that

\displaystyle  f_0(a) \approx (-1)^k f_0(ap).

But if {k} is odd, one can use Dirichlet’s theorem on primes in arithmetic progressions to restrict to primes {p} that are {1} modulo the period of {f_0}, and conclude now that {f_0} 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 {p} using the “{W}-trick”, as well as known facts about the Gowers uniformity of the von Mangoldt function, one can obtain an approximation of the form

\displaystyle  (-1)^k f(a) \approx {\bf E}_{b: (b,W)=1} f(ab)

where {b} ranges over a large range of integers coprime to some primorial {W = \prod_{p \leq w} p}. On the other hand, by iterating (4) we have

\displaystyle  f(a) \approx f(apq)

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

\displaystyle  f(a) \approx {\bf E}_{b: (b,W)=1} f(ab).

For {k} odd, one can combine the two approximations to conclude that {f(a)=0}. (This argument is not given in the current paper, but we plan to detail it in a subsequent one.)

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