On a Ramanujan type expansion of arithmetical functions

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.