Some asymptotic formulae involving Cohen-Ramanujan expansions

Abstract

Some necessary and sufficient conditions for the existence of Cohen-Ramanujan expansions for arithmetical functions were provided by these authors in [\textit{arXive preprint arXive:2205.08466}, 2022]. Given two arithmetical functions $f$ and $g$ with absolutely convergent Cohen-Ramanujan expansions, we derive an asymptotic formula for ${\sum }_{n\le N}f(n)g(n+h)$ where $h$ is a fixed positive integer. We also provide Cohen-Ramanujan expansions for certain functions to illustrate some of the results we prove consequently.