An averaged form of Chowla's conjecture

DOI10.2140/ANT.2015.9.2167zbMATH Open1377.11109arXiv1503.05121OpenAlexW1557923169WikidataQ122906092 ScholiaQ122906092MaRDI QIDQ901784

Author name not available (Why is that?)

Publication date: 12 January 2016

Published in: (Search for Journal in Brave)

Abstract: Let $lambda$ denote the Liouville function. A well known conjecture of Chowla asserts that for any distinct natural numbers $h_1,dots,h_k$, one has $su{m}_{1leqnleqX}lambda(n+h_1)dotsmlambda(n+h_k)=o(X)$ as $Xoinfty$. This conjecture remains unproven for any $h_1,dots,h_k$ with $kgeq2$. In this paper, using the recent results of the first two authors on mean values of multiplicative functions in short intervals, combined with an argument of Katai and Bourgain-Sarnak-Ziegler, we establish an averaged version of this conjecture, namely ${\displaystyle }$sum_{h_1,dots,h_k leq H} left|sum_{1 leq n leq X} lambda(n+h_1) dotsm lambda(n+h_k) ight| = o(H^kX)${\displaystyle }$ as $Xoinfty$ whenever $H=H(X)leqX$ goes to infinity as $Xoinfty$, and $k$ is fixed. Related to this, we give the exponential sum estimate ${\displaystyle }$ int_0^X left|sum_{x leq n leq x+H} lambda(n) e(alpha n) ight| dx = o( HX )${\displaystyle }$ as $Xoinfty$ uniformly for all $alphainmathbbR$, with $H$ as before. Our arguments in fact give quantitative bounds on the decay rate (roughly on the order of $fracloglogHlogH$), and extend to more general bounded multiplicative functions than the Liouville function, yielding an averaged form of a (corrected) conjecture of Elliott.

Full work available at URL: https://arxiv.org/abs/1503.05121

zbMATH Keywords

No records found.

Mathematics Subject Classification ID

No records found.

This page was built for publication: An averaged form of Chowla's conjecture

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q901784)