Finite-part integral representation of the Riemann zeta function at odd positive integers and consequent representations

2022/03/21 by Eric A. Galapon, Galapon, Eric A.
Mathematics · Physics and Astronomy · #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #Analytic Number Theory Research #Complex Variables (math.CV) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2203.11342

openalex publication_date 2022/03/21 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28

Abstract

The values of the Riemann zeta function at odd positive integers, ζ(2n+1), are shown to admit a representation proportional to the finite-part of the divergent integral ∫₀^∞ t^-2n-1 cscht dt. Integral representations for ζ(2n+1) are then deduced from the finite-part integral representation. Certain relations between ζ(2n+1) and ζ'(2n+1) are likewise deduced, from which integral representations for ζ'(2n+1) are obtained.

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