Two identities involving Cohen-Ramanujan expansions

Arya Chandran; K Vishnu Namboothiri

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arXiv · 2503.12027

Two identities involving Cohen-Ramanujan expansions

Arya Chandran·K Vishnu Namboothiri

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Abstract

An arithmetical function $f$ is said to admit a \emph{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 $\widehat{f}(r)$. 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 $\sum\limits_{n \leq N} \frac{J_a(n)}{n^a} \frac{J_b(n+h)}{(n+h)^b}$ using the expansion we derive.

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Arya Chandran, K Vishnu Namboothiri. 2025-03-15. Two identities involving Cohen-Ramanujan expansions. https://arxiv.org/abs/2503.12027

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