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 $f(n)=\sum _{k=1}^{\infty }a(k)c_k(n)$ where $a(k)$ are complex numbers and $c_k(n):=\sum _{\begin{array}{c}m=1\\ (m,k)=1\end{array}}^k{e}^{\frac{2\pi imn}{k}}$ is the Ramanujan sum. Here we prove a reciprocity result on Cohen-Ramanujan sums $c_k^s(n):=\sum _{\begin{array}{c}h=1\\ (h,k^s{)}_s=1\end{array}}^{k^s}{e}^{\frac{2\pi inh}{k^s}}$ 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 $\sum _{k=1}^{\infty }a(k)c_k^{(s)}(n)$ exist if and only if expansions in the form $\sum _{k=1}^{\infty }b(k/n)c_n^{(s)}(k)$ exist.