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

TL;DR

This paper improves asymptotic formulas with error terms for convolution sums of arithmetical functions with Ramanujan expansions, under weaker hypotheses, advancing the understanding of Ramanujan expansions in number theory.

Contribution

It weakens the conditions needed to derive asymptotic formulas with error terms for Ramanujan expansion convolutions, extending previous results.

Findings

Enhanced asymptotic formulas with error bounds under minimal hypotheses

Broader applicability to arithmetical functions with Ramanujan expansions

Refined understanding of convolution sums in number theory

Abstract

For two arithmetical functions $f$ and $g$ with absolutely convergent Ramanujan expansions, Murty and Saha have recently derived asymptotic formulas with error term for the convolution sum $\sum_{n \le N} f(n) g(n+h)$ under some suitable conditions (see http://arxiv.org/abs/1506.01945). In this follow up article we improve these results with a weakened hypothesis which is in some sense minimal.