A reciprocity theorem for the Cohen-Ramanujan sums and its application to Cohen-Ramanujan expansions in the second variable

Abstract

For an arithmetical function $f$, its Ramanujan expansion is a series expansion in the form $f(n)=\sum\limits_{k=1}^{\infty}a(k) c_k(n)$ where $a(k)$ are complex numbers and $c_k(n):= \sum\limits_{\substack{m=1\\(m, k)=1}}^{k}e^{\frac{2\pi imn}{k}}$ is the Ramanujan sum. Here we prove a reciprocity result on Cohen-Ramanujan sums $c_k^s(n) :=\sum\limits_{\substack{{h=1}\\(h,k^s)_s=1}}^{k^s}e^{\frac{2\pi i n h}{k^s}}$ to change the position of $k$ and $n$ in a twisted function and use it to prove that for certain arithmetical functions $f$, Cohen-Ramanujan series expansions in the form $\sum\limits_{k=1}^{\infty}a(k) c_k^{(s)}(n)$ exist if and only if expansions in the form $\sum\limits_{k=1}^{\infty}b(k/n) c_n^{(s)}(k)$ exist.