On correlations of certain multiplicative functions

TL;DR

This paper investigates the asymptotic behavior of sums involving shifted products of multiplicative functions, providing new formulas and improved error estimates that are independent of the shift parameter.

Contribution

It derives asymptotic formulas for sums of shifted products of multiplicative functions, including cases with the Möbius squared function, with enhanced error bounds.

Findings

Asymptotic formulas for sums of shifted products of multiplicative functions.

Improved error terms that are independent of the shift parameter.

Extensions to sums involving the Möbius squared function.

Abstract

In this paper, we study sums of shifted products $\sum\limits_{n \leq x} F(n) G(n-h)$ for any $|h| \leq x/2$ and arithmetic functions $F=f*1$ and $G=g*1$, with $f$ and $g$ small. We obtain asymptotic formula for different orders of magnitude of $f$ and $g$. We also provide asymptotic formula for sums of the type $\sum\limits_{n \leq x} \mu^2(n) G(n-h)$, where $G=g*1$ and $g$ is small. For small order of magnitudes of $f$ and $g$, we improve the error terms and make them independent of $h$.