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

2023/12/10 by Sivadasan, Vinod, Namboothiri, K Vishnu
#11A25 #11L03 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2312.05936

Abstract

Srinivasa Ramanujan provided Fourier series expansions of certain arithmetical functions in terms of the exponential sum defined by c_q(n)=∑_\substackm=1 (m,q)=1^qe^{(2 πimn)/(q)}. Later, H. Delange derived the bound ∑_q|k|c_q(n)|≤ n 2^ω(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 [Duke Mathematical Journal, 16(85-90):2, 1949]. We derive a bound as well as exact value for ∑_q|k|c_q^(s)(n)| and provide a sufficient condition for the Cohen-Ramanujan expansions to exist.

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