Some asymptotic formulae involving Cohen-Ramanujan expansions

TL;DR

This paper develops asymptotic formulas for sums involving products of arithmetical functions expanded via Cohen-Ramanujan sums, extending Ramanujan sum techniques and providing explicit expansions for specific functions.

Contribution

It introduces asymptotic formulas for sums of products of functions with Cohen-Ramanujan expansions and offers explicit expansions for particular functions.

Findings

Derived asymptotic formula for sums of f(n)g(n+h) with Cohen-Ramanujan expansions

Provided explicit Cohen-Ramanujan expansions for specific arithmetical functions

Extended Ramanujan sum techniques to Cohen-Ramanujan sums

Abstract

Cohen-Ramanujan sum, denoted by $c_r^s(n)$, is an exponential sum similar to the Ramanujan sum $c_r(n):=\sum\limits_{\substack{h=1\\{(h,r)=1}}}^{r}e^{\frac{2\pi i n h}{r}}$. An arithmetical function $f$ is said to admit a Cohen-Ramanujan expansion $ f(n):=\sum\limits_{r}\widehat{f}(r)c_r^s(n)$ if the series on the right hand side converges for suitable complex numbers $\widehat{f}(r)$. Given two arithmetical functions $f$ and $g$ with absolutely convergent Cohen-Ramanujan expansions, we derive an asymptotic formula for the sum $\sum\limits_{\substack{n\leq N}}f(n)g(n+h)$ where $h$ is a fixed non negative integer. We also provide Cohen-Ramanujan expansions for certain functions to illustrate some of the results we prove consequently.