On the symmetry of the Liouville function in almost all short intervals

TL;DR

The paper establishes that the Liouville function exhibits near-symmetry in most short intervals, providing bounds for its symmetry integral and extending results to related functions like the Möbius function.

Contribution

It introduces non-trivial bounds for the symmetry integral of the Liouville function in short intervals and extends these results to related arithmetic functions.

Findings

Bound for the symmetry integral: $I_{\lambda}(N,h) \\ll NhL^3+Nh^{21/20}$

Results apply to the Möbius function $\\mu(n)$ on square-free integers

Demonstrates near-symmetry of the Liouville function in almost all short intervals

Abstract

We prove a kind of "almost all symmetry" result for the Liouville function $\lambda(n):=(-1)^{\Omega(n)}$, giving non-trivial bounds for its "symmetry integral", say $I_{\lambda}(N,h)$ : we get $I_{\lambda}(N,h)\ll NhL^3+Nh^{21/20}$, with $L:=\log N$. We also give similar results for other related arithmetic functions, like the M\"{o}bius function $\mu(n)$ ($=\lambda(n)$ on square-free $n$).