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 $c_r(n)=\sum _{\begin{array}{c}m=1\\ (m,r)=1\end{array}}^r{e}^{\frac{2\pi imn}{r}}$ 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 $c_r^s(n)=\sum _{\begin{array}{c}h=1\\ (h,r^s{)}_s=1\end{array}}^{r^s}{e}^{\frac{2\pi inh}{r^s}}$. We also provide some necessary and sufficient conditions for such expansions to exist.

arXiv commentThe techniques which we adopted here can be used to derive infinite series expansions of some arithmetical functions using some other generalizations of Ramanujan sum