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arXiv · 2312.05936

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

Vinod Sivadasan·K Vishnu Namboothiri
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Abstract

Srinivasa Ramanujan provided Fourier series expansions of certain arithmetical 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 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 $\sum\limits_{q|k}|c_q^{(s)}(n)|$ and provide a sufficient condition for the Cohen-Ramanujan expansions to exist.

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Vinod Sivadasan, K Vishnu Namboothiri. 2023-12-10. On an identity of Delange and its application to Cohen-Ramanujan expansions. https://arxiv.org/abs/2312.05936

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