Autocorrelation of the Mobius Function

TL;DR

This paper establishes an effective asymptotic bound for the autocorrelation of the Mobius function, showing it diminishes rapidly as x increases, which advances understanding of its randomness properties.

Contribution

It provides a new effective asymptotic estimate for the autocorrelation of the Mobius function, improving previous bounds and insights into its distribution.

Findings

Autocorrelation sum is bounded by an exponential decay in ^{rac{1}{2}\

,

,

Abstract

Let $x\geq 1$ be a large integer, and let $\mu:\mathbb{N}\longrightarrow\{-1,0,1\}$ be the Mobius function. This article proposes an effective asymptotic result for the autocorrelation function $\sum_{n \leq x} \mu(n) \mu(n+t) =O\left( e^{-c\sqrt{\log x}}\right) $, where $t\ne 0$ be a small fixed integer, and $c>0$ is a constant.