On a Ramanujan type expansion of arithmetical functions

TL;DR

This paper extends Ramanujan's series expansions of arithmetical functions by using Cohen-Ramanujan sums, providing conditions for their existence and exploring their properties.

Contribution

It introduces a new type of expansion for arithmetical functions using Cohen-Ramanujan sums and establishes criteria for their validity.

Findings

Derived expansions in terms of Cohen-Ramanujan sums.

Provided necessary and sufficient conditions for these expansions.

Connected classical Ramanujan sums with Cohen's generalization.

Abstract

Srinivasa Ramanujan provided series expansions of certain arithmetical functions in terms of the exponential sums defined by $c_r(n) = \sum\limits_{\substack{{m=1}\\ (m,r)=1}}^{r} e^{\frac{2 \pi imn}{r}}$ in [Trans. Cambridge Phillos. Soc, 22(13):259-276,1918]. Here we give similar type of expansions in terms of the Cohen-Ramanujan sum defined by E. Cohen in [Duke Mathematical Journal, 16(85-90):2, 1949] as $c_r^s(n)=\sum\limits_{\substack{h=1\\ (h,r^s)_s=1}}^{r^s}e^{\frac{2\pi i n h}{r^s}}$. We also provide some necessary and sufficient conditions for such expansions to exist.