On an identity of Delange and its application to Cohen-Ramanujan expansions

Srinivasa Ramanujan provided Fourier series expansions of certain arithmetical functions in terms of the exponential sum defined by $c_q(n)={\sum }_{\begin{array}{c}m=1\\ (m,q)=1\end{array}}^q{e}^{\frac{2\pi imn}{q}}$. Later, H. Delange derived the bound $\sum _{q\mathrm{\mid }k}\mathrm{\mid }{c}_q(n)\mathrm{\mid }\le n{2}^{\omega (k)}$ and gave a sufficient condition for such expansions to exist. A. Grytczuk gave an exact value for this bound, and derived a converse implication of the absolute convergence stated by H. Delange. We here show that these results have natural generalizations in terms of the Cohen-Ramanujan sum $c_q^{(s)}(n)$ defined by E. Cohen in [\emph{Duke Mathematical Journal, 16(85-90):2, 1949}]. We derive a bound as well as exact value for $\sum _{q\mathrm{\mid }k}\mathrm{\mid }{c}_q^{(s)}(n)\mathrm{\mid }$ and provide a sufficient condition for the Cohen-Ramanujan expansions to exist.