The logarithmically averaged Chowla and Elliott conjectures for two-point correlations

Abstract

Let $\lambda$ denote the Liouville function. The Chowla conjecture, in the two-point correlation case, asserts that $$\sum _{n\le x}\lambda (a_{1}n+b_1)\lambda (a_{2}n+b_2)=o(x)$$ as $x\to \infty$, for any fixed natural numbers $a_1,a_2,b_1,b_2$ with $a_1{b}_2-a_2{b}_1\ne 0$. In this paper we establish the logarithmically averaged version $$\sum _{x/\omega (x)<n\le x}\frac{\lambda (a_{1}n+b_1)\lambda (a_{2}n+b_2)}{n}=o(\log \omega (x))$$ of the Chowla conjecture as $x\to \infty$, where $1\le \omega (x)\le x$ is an arbitrary function of $x$ that goes to infinity as $x\to \infty$, thus breaking the "parity barrier" for this problem. Our main tools are the multiplicativity of the Liouville function at small primes, a recent result of Matomäki, Radziwiłł, and the author on the averages of modulated multiplicative functions in short intervals, concentration of measure inequalities, the Hardy-Littlewood circle method combined with a restriction theorem for the primes, and a novel "entropy decrement argument". Most of these ingredients are also available (in principle, at least) for the higher order correlations, with the main missing ingredient being the need to control short sums of multiplicative functions modulated by local nilsequences. Our arguments also extend to more general bounded multiplicative functions than the Liouville function $\lambda$, leading to a logarithmically averaged version of the Elliott conjecture in the two-point case. In a subsequent paper we will use this version of the Elliott conjecture to affirmatively settle the Erdős discrepancy problem.

32 pages, no figures, submitted, Forum of Mathematics, Pi. This is the final version, incorporating the second round of referee comments and also with updated references

References in corpus (4)

• An averaged form of Chowla's conjecture
• Asymptotics for multilinear averages of multiplicative functions
• Sign patterns of the Liouville and Möbius functions
• The Erdos discrepancy problem

Cited by in corpus (7)

• Odd order cases of the logarithmically averaged Chowla conjecture
• Correlations of multiplicative functions in function fields
• Effective Asymptotic Formulae for Multilinear Averages of Multiplicative Functions
• The Erdos discrepancy problem
• The Liouville function in short intervals [after Matomaki and Radziwill]
• Harmonic Analysis on the Positive Rationals. Determination of the Group Generated by the Ratios $(an+b)/(An+B)$
• Random Chowla's Conjecture for Rademacher Multiplicative Functions