Some asymptotic formulae involving Cohen-Ramanujan expansions

2024/11/07 by Chandran, Arya, K, Vishnu Namboothiri
#11A25 #11L03 #11N05 #11N37 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2411.11890

Abstract

Cohen-Ramanujan sum, denoted by c_r^s(n), is an exponential sum similar to the Ramanujan sum c_r(n):=∑_\substackh=1 (h,r)=1^re^{(2πi n h)/(r)}. An arithmetical function f is said to admit a Cohen-Ramanujan expansion f(n):=∑_r\widehatf(r)c_r^s(n) if the series on the right hand side converges for suitable complex numbers \widehatf(r). Given two arithmetical functions f and g with absolutely convergent Cohen-Ramanujan expansions, we derive an asymptotic formula for the sum ∑_\substackn≤ Nf(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.

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