On a Ramanujan type expansion of arithmetical functions

2022/05/17 by Arya Chandran, Chandran, Arya, K Vishnu Namboothiri +1
Mathematics · #11A25 #11L03 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2205.08466

openalex publication_date 2022/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Srinivasa Ramanujan provided series expansions of certain arithmetical functions in terms of the exponential sums defined by c_r(n) = ∑_\substackm=1 (m,r)=1^r e^{(2 πimn)/(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)=∑_\substackh=1 (h,r^s)_s=1^rse^{(2πi n h)/(rs)}. We also provide some necessary and sufficient conditions for such expansions to exist.

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