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Srinivasa Ramanujan provided Fourier series expansions of certainarithmetical functions in terms of the exponential sum defined by$c_q(n)=\sum\limits_{\substack{{m=1}\\(m,q)=1}}^{q}e^{\frac{2 \pi imn}{q}}$.Later, H. Delange derived the bound $\sum\limits_{q|k}|c_q(n)|\leq n\,2^{\omega(k)}$ and gave a sufficient condition for such expansions to exist. A.Grytczuk gave an exact value for this bound, and derived a converse implicationof the absolute convergence stated by H. Delange. We here show that theseresults 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$\sum\limits_{q|k}|c_q^{(s)}(n)|$ and provide a sufficient condition for theCohen-Ramanujan expansions to exist.

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