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Kaisa Matomäki, Maksym Radziwill, Joni Teräväinen, Tamar Ziegler and I have uploaded to the arXiv our paper Higher uniformity of bounded multiplicative functions in short intervals on average. This paper (which originated from a working group at an AIM workshop on Sarnak’s conjecture) focuses on the local Fourier uniformity conjecture for bounded multiplicative functions such as the Liouville function ${\lambda}$. One form of this conjecture is the assertion that

$\displaystyle \int_0^X \| \lambda \|_{U^k([x,x+H])}\ dx = o(X) \ \ \ \ \ (1)$

as ${X \rightarrow \infty}$ for any fixed ${k \geq 0}$ and any ${H = H(X) \leq X}$ that goes to infinity as ${X \rightarrow \infty}$, where ${U^k([x,x+H])}$ is the (normalized) Gowers uniformity norm. Among other things this conjecture implies (logarithmically averaged version of) the Chowla and Sarnak conjectures for the Liouville function (or the Möbius function), see this previous blog post.

The conjecture gets more difficult as ${k}$ increases, and also becomes more difficult the more slowly ${H}$ grows with ${X}$. The ${k=0}$ conjecture is equivalent to the assertion

$\displaystyle \int_0^X |\sum_{x \leq n \leq x+H} \lambda(n)| \ dx = o(HX)$

which was proven (for arbitrarily slowly growing ${H}$) in a landmark paper of Matomäki and Radziwill, discussed for instance in this blog post.

For ${k=1}$, the conjecture is equivalent to the assertion

$\displaystyle \int_0^X \sup_\alpha |\sum_{x \leq n \leq x+H} \lambda(n) e(-\alpha n)| \ dx = o(HX). \ \ \ \ \ (2)$

This remains open for sufficiently slowly growing ${H}$ (and it would be a major breakthrough in particular if one could obtain this bound for ${H}$ as small as ${\log^\varepsilon X}$ for any fixed ${\varepsilon>0}$, particularly if applicable to more general bounded multiplicative functions than ${\lambda}$, as this would have new implications for a generalization of the Chowla conjecture known as the Elliott conjecture). Recently, Kaisa, Maks and myself were able to establish this conjecture in the range ${H \geq X^\varepsilon}$ (in fact we have since worked out in the current paper that we can get ${H}$ as small as ${\exp(\log^{5/8+\varepsilon} X)}$). In our current paper we establish Fourier uniformity conjecture for higher ${k}$ for the same range of ${H}$. This in particular implies local orthogonality to polynomial phases,

$\displaystyle \int_0^X \sup_{P \in \mathrm{Poly}_{\leq k-1}({\bf R} \rightarrow {\bf R})} |\sum_{x \leq n \leq x+H} \lambda(n) e(-P(n))| \ dx = o(HX) \ \ \ \ \ (3)$

where ${\mathrm{Poly}_{\leq k-1}({\bf R} \rightarrow {\bf R})}$ denotes the polynomials of degree at most ${k-1}$, but the full conjecture is a bit stronger than this, establishing the more general statement

$\displaystyle \int_0^X \sup_{g \in \mathrm{Poly}({\bf R} \rightarrow G)} |\sum_{x \leq n \leq x+H} \lambda(n) \overline{F}(g(n) \Gamma)| \ dx = o(HX) \ \ \ \ \ (4)$

for any degree ${k}$ filtered nilmanifold ${G/\Gamma}$ and Lipschitz function ${F: G/\Gamma \rightarrow {\bf C}}$, where ${g}$ now ranges over polynomial maps from ${{\bf R}}$ to ${G}$. The method of proof follows the same general strategy as in the previous paper with Kaisa and Maks. (The equivalence of (4) and (1) follows from the inverse conjecture for the Gowers norms, proven in this paper.) We quickly sketch first the proof of (3), using very informal language to avoid many technicalities regarding the precise quantitative form of various estimates. If the estimate (3) fails, then we have the correlation estimate

$\displaystyle |\sum_{x \leq n \leq x+H} \lambda(n) e(-P_x(n))| \gg H$

for many ${x \sim X}$ and some polynomial ${P_x}$ depending on ${x}$. The difficulty here is to understand how ${P_x}$ can depend on ${x}$. We write the above correlation estimate more suggestively as

$\displaystyle \lambda(n) \sim_{[x,x+H]} e(P_x(n)).$

Because of the multiplicativity ${\lambda(np) = -\lambda(p)}$ at small primes ${p}$, one expects to have a relation of the form

$\displaystyle e(P_{x'}(p'n)) \sim_{[x/p,x/p+H/p]} e(P_x(pn)) \ \ \ \ \ (5)$

for many ${x,x'}$ for which ${x/p \approx x'/p'}$ for some small primes ${p,p'}$. (This can be formalised using an inequality of Elliott related to the Turan-Kubilius theorem.) This gives a relationship between ${P_x}$ and ${P_{x'}}$ for “edges” ${x,x'}$ in a rather sparse “graph” connecting the elements of say ${[X/2,X]}$. Using some graph theory one can locate some non-trivial “cycles” in this graph that eventually lead (in conjunction to a certain technical but important “Chinese remainder theorem” step to modify the ${P_x}$ to eliminate a rather serious “aliasing” issue that was already discussed in this previous post) to obtain functional equations of the form

$\displaystyle P_x(a_x \cdot) \approx P_x(b_x \cdot)$

for some large and close (but not identical) integers ${a_x,b_x}$, where ${\approx}$ should be viewed as a first approximation (ignoring a certain “profinite” or “major arc” term for simplicity) as “differing by a slowly varying polynomial” and the polynomials ${P_x}$ should now be viewed as taking values on the reals rather than the integers. This functional equation can be solved to obtain a relation of the form

$\displaystyle P_x(t) \approx T_x \log t$

for some real number ${T_x}$ of polynomial size, and with further analysis of the relation (5) one can make ${T_x}$ basically independent of ${x}$. This simplifies (3) to something like

$\displaystyle \int_0^X \sup_{P \in \mathrm{Poly}_{\leq k-1}({\bf R} \rightarrow {\bf R})} |\sum_{x \leq n \leq x+H} \lambda(n) n^{-iT}| \ dx = o(HX)$

and this is now of a form that can be treated by the theorem of Matomäki and Radziwill (because ${n \mapsto \lambda(n) n^{-iT}}$ is a bounded multiplicative function). (Actually because of the profinite term mentioned previously, one also has to insert a Dirichlet character of bounded conductor into this latter conclusion, but we will ignore this technicality.)

Now we apply the same strategy to (4). For abelian ${G}$ the claim follows easily from (3), so we focus on the non-abelian case. One now has a polynomial sequence ${g_x \in \mathrm{Poly}({\bf R} \rightarrow G)}$ attached to many ${x \sim X}$, and after a somewhat complicated adaptation of the above arguments one again ends up with an approximate functional equation

$\displaystyle g_x(a_x \cdot) \Gamma \approx g_x(b_x \cdot) \Gamma \ \ \ \ \ (6)$

where the relation ${\approx}$ is rather technical and will not be detailed here. A new difficulty arises in that there are some unwanted solutions to this equation, such as

$\displaystyle g_x(t) = \gamma^{\frac{\log(a_x t)}{\log(a_x/b_x)}}$

for some ${\gamma \in \Gamma}$, which do not necessarily lead to multiplicative characters like ${n^{-iT}}$ as in the polynomial case, but instead to some unfriendly looking “generalized multiplicative characters” (think of ${e(\lfloor \alpha \log n \rfloor \beta \log n)}$ as a rough caricature). To avoid this problem, we rework the graph theory portion of the argument to produce not just one functional equation of the form (6)for each ${x}$, but many, leading to dilation invariances

$\displaystyle g_x((1+\theta) t) \Gamma \approx g_x(t) \Gamma$

for a “dense” set of ${\theta}$. From a certain amount of Lie algebra theory (ultimately arising from an understanding of the behaviour of the exponential map on nilpotent matrices, and exploiting the hypothesis that ${G}$ is non-abelian) one can conclude that (after some initial preparations to avoid degenerate cases) ${g_x(t)}$ must behave like ${\gamma_x^{\log t}}$ for some central element ${\gamma_x}$ of ${G}$. This eventually brings one back to the multiplicative characters ${n^{-iT}}$ that arose in the polynomial case, and the arguments now proceed as before.

We give two applications of this higher order Fourier uniformity. One regards the growth of the number

$\displaystyle s(k) := |\{ (\lambda(n+1),\dots,\lambda(n+k)): n \in {\bf N} \}|$

of length ${k}$ sign patterns in the Liouville function. The Chowla conjecture implies that ${s(k) = 2^k}$, but even the weaker conjecture of Sarnak that ${s(k) \gg (1+\varepsilon)^k}$ for some ${\varepsilon>0}$ remains open. Until recently, the best asymptotic lower bound on ${s(k)}$ was ${s(k) \gg k^2}$, due to McNamara; with our result, we can now show ${s(k) \gg_A k^A}$ for any ${A}$ (in fact we can get ${s(k) \gg_\varepsilon \exp(\log^{8/5-\varepsilon} k)}$ for any ${\varepsilon>0}$). The idea is to repeat the now-standard argument to exploit multiplicativity at small primes to deduce Chowla-type conjectures from Fourier uniformity conjectures, noting that the Chowla conjecture would give all the sign patterns one could hope for. The usual argument here uses the “entropy decrement argument” to eliminate a certain error term (involving the large but mean zero factor ${p 1_{p|n}-1}$). However the observation is that if there are extremely few sign patterns of length ${k}$, then the entropy decrement argument is unnecessary (there isn’t much entropy to begin with), and a more low-tech moment method argument (similar to the derivation of Chowla’s conjecture from Sarnak’s conjecture, as discussed for instance in this post) gives enough of Chowla’s conjecture to produce plenty of length ${k}$ sign patterns. If there are not extremely few sign patterns of length ${k}$ then we are done anyway. One quirk of this argument is that the sign patterns it produces may only appear exactly once; in contrast with preceding arguments, we were not able to produce a large number of sign patterns that each occur infinitely often.

The second application is to obtain cancellation for various polynomial averages involving the Liouville function ${\lambda}$ or von Mangoldt function ${\Lambda}$, such as

$\displaystyle {\bf E}_{n \leq X} {\bf E}_{m \leq X^{1/d}} \lambda(n+P_1(m)) \lambda(n+P_2(m)) \dots \lambda(n+P_k(m))$

or

$\displaystyle {\bf E}_{n \leq X} {\bf E}_{m \leq X^{1/d}} \lambda(n+P_1(m)) \Lambda(n+P_2(m)) \dots \Lambda(n+P_k(m))$

where ${P_1,\dots,P_k}$ are polynomials of degree at most ${d}$, no two of which differ by a constant (the latter is essential to avoid having to establish the Chowla or Hardy-Littlewood conjectures, which of course remain open). Results of this type were previously obtained by Tamar Ziegler and myself in the “true complexity zero” case when the polynomials ${P}$ had distinct degrees, in which one could use the ${k=0}$ theory of Matomäki and Radziwill; now that higher ${k}$ is available at the scale ${H=X^{1/d}}$ we can now remove this restriction.

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 ${\lambda}$. For instance, with regards to length 5 sign patterns

$\displaystyle (\lambda(n+1),\dots,\lambda(n+5)) \in \{-1,+1\}^5$

of the Liouville function, we can now show that at least ${24}$ of the ${32}$ possible sign patterns in ${\{-1,+1\}^5}$ occur with positive upper density. (Conjecturally, all of them do so, and this is known for all shorter sign patterns, but unfortunately ${24}$ seems to be the limitation of our methods.)

The Liouville function can be written as ${\lambda(n) = e^{2\pi i \Omega(n)/2}}$, where ${\Omega(n)}$ is the number of prime factors of ${n}$ (counting multiplicity). One can also consider the variant ${\lambda_3(n) = e^{2\pi i \Omega(n)/3}}$, which is a completely multiplicative function taking values in the cube roots of unity ${\{1, \omega, \omega^2\}}$. Here we are able to show that all ${27}$ sign patterns in ${\{1,\omega,\omega^2\}}$ occur with positive lower density as sign patterns ${(\lambda_3(n+1), \lambda_3(n+2), \lambda_3(n+3))}$ of this function. The analogous result for ${\lambda}$ 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 ${1/8}$ (from this paper of myself and Teräväinen), but these techniques barely fail to handle the ${\lambda_3}$ case by itself (largely because the “parity” arguments used in the case of the Liouville function no longer control three-point correlations in the ${\lambda_3}$ 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 ${a \in A_1, a+r \in A_2, a+2r \in A_3}$ for a certain partition ${G = A_1 \cup A_2 \cup A_3}$ of a compact abelian group ${G}$ (think for instance of the unit circle ${G={\bf R}/{\bf Z}}$, although the general case is a bit more complicated, in particular if ${G}$ is disconnected then there is a certain “coprimality” constraint on ${r}$, also we can allow the ${A_1,A_2,A_3}$ to be replaced by any ${A_{c_1}, A_{c_2}, A_{c_3}}$ with ${c_1+c_2+c_3}$ divisible by ${3}$), with each of the ${A_i}$ having measure ${1/3}$. 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 ${P^+(n)}$ of a natural number ${n}$. For instance, we improve results of Erdös-Pomerance and of Balog demonstrating that the inequalities

$\displaystyle P^+(n+1) < P^+(n+2) < P^+(n+3)$

and

$\displaystyle P^+(n+1) > P^+(n+2) > P^+(n+3)$

each hold for infinitely many ${n}$, by demonstrating the stronger claims that the inequalities

$\displaystyle P^+(n+1) < P^+(n+2) < P^+(n+3) > P^+(n+4)$

and

$\displaystyle P^+(n+1) > P^+(n+2) > P^+(n+3) < P^+(n+4)$

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

$\displaystyle P^+(n+1), P^+(n+2), P^+(n+3) > n^\gamma$

for any fixed ${\gamma < e^{-1/3} = 0.7165\dots}$ (this improves on a previous result of Hildebrand with ${e^{-1/3}}$ replaced by ${e^{-1/2} = 0.6065\dots}$. 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 ${A}$ which have some multiplicative structure, in the sense that (roughly speaking) there is a set ${B}$ such that for all small primes ${p}$, the statements ${n \in A}$ and ${pn \in B}$ are roughly equivalent to each other. For instance, if ${A}$ is a level set ${A = \{ n: \omega(n) = 0 \hbox{ mod } 3 \}}$, one would take ${B = \{ n: \omega(n) = 1 \hbox{ mod } 3 \}}$; if instead ${A}$ is a set of the form ${\{ n: P^+(n) \geq n^\gamma\}}$, then one can take ${B=A}$. 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

$\displaystyle {\bf E}_n 1_A(n+1) 1_A(n+2) 1_A(n+3)$

with a two-parameter correlation such as

$\displaystyle {\bf E}_n {\bf E}_p 1_B(n+p) 1_B(n+2p) 1_B(n+3p)$

(where we will be deliberately vague as to how we are averaging over ${n}$ and ${p}$), 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

$\displaystyle {\bf E}_n {\bf E}_r 1_B(n+r) 1_B(n+2r) 1_B(n+3r)$

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

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

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

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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