Two identities involving Cohen-Ramanujan expansions

2025/03/15 by Chandran, Arya, Namboothiri, K Vishnu
#11A25 #11L03 #11N05 #11N37 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2503.12027

Abstract

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). Here c_r^s(n) denotes the Cohen-Ramanujan sum defined by E. Cohen. We deduce here a Cohen-Ramanujan expansion for the Jordan totient function J_k(n). Further, we give an an asymptotic formula for the sum ∑_{n ≤ N} (J_a(n))/(n^a) (J_b(n+h))/((n+h)^b) using the expansion we derive.

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