On the error term in a Parseval type formula in the theory of Ramanujan expansions

TL;DR

This paper develops asymptotic formulas with error estimates for convolution sums of arithmetical functions using Ramanujan expansions, extending previous work in the field.

Contribution

It introduces a new approach leveraging Ramanujan expansions to analyze convolution sums with error terms, advancing the understanding of their asymptotic behavior.

Findings

Derived asymptotic formulas with error terms for convolution sums

Extended previous results by Gadiyar, Murty, and Padma

Applied Ramanujan expansions to obtain precise estimates

Abstract

Given two arithmetical functions $f,g$ we derive, under suitable conditions, asymptotic formulas with error term, for the convolution sums $\sum_{n \le N} f(n) g(n+h)$, building on an earlier work of Gadiyar, Murty and Padma. A key role in our method is played by the theory of Ramanujan expansions for arithmetical functions.