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

Abstract

We study the asymptotic behaviour of higher order correlations E n≤X/d g 1 (n + ah 1 ) · · · g k (n + ah k ) as a function of the parameters a and d, where g 1 , . . . , g k are bounded multiplicative functions, h 1 , . . . , h k are integer shifts, and X is large. Our main structural result asserts, roughly speaking, that such correlations asymptotically vanish for almost all X if g 1 · · · g k does not (weakly) pretend to be a twisted Dirichlet character n → χ(n)n it , and behave asymptotically like a multiple of d −it χ(a) otherwise. This extends our earlier work on the structure of logarithmically averaged correlations, in which the d parameter is averaged out and one can set t = 0. Among other things, the result enables us to establish special cases of the Chowla and Elliott conjectures for (unweighted) averages at almost all scales; for instance, we establish the k-point Chowla conjecture E n≤X λ(n + h 1 ) · · · λ(n + h k ) = o(1) for k odd or equal to 2 for all scales X outside of a set of zero logarithmic density.

Correlations of multiplicative functions along deterministic and independent sequences

“…, a ℓ (n) are distinct. Hence, Theorem 4.7 applies and shows that in proving (27) we can assume that the system we work with is (Z k , µ k , T ) for some k ∈ N. We conclude the reduction as in the previous proposition. Proof.…”

“…Recently, a version involving logarithmic averages was established for ℓ = 2 by Tao [23] and for all odd values of ℓ by Tao and Teräväinen [25,26]. Similar results are also known for Cesàro averages on almost all scales [27]. But even in its logarithmic form, the conjecture remains open for all even ℓ ≥ 4.…”

“…If not, as before, we can assume that k 1 = 0. Then the right hand side in (27) is 0 and the left hand side is equal to…”

Rigidity in dynamics and M

“…We remark that Corollary 1.5 is related to a recent result of Tao and Teräväinen [26] according to which Chowla's conjecture holds for "most" subsequences. However, given a particular subsequence {N n k } their result cannot guarantee the existence of even one subsequence of {N n k } along which Chowla would hold.…”

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

“…. , h k , the ordinary Chowla conjecture (1) implies its logarithmically averaged counterpart (3); however there does not seem to be any easy way to reverse this implication 2 ; see however [43].…”