Integral representations of the Riemann zeta function of odd argument

2024/10/30 by Jean‐Christophe Pain, Pain, Jean-Christophe
Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2410.23096

openalex publication_date 2024/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article we obtain, using an expression of the digamma function ψ(x) due to Mikolas, integral representations of the zeta function of odd arguments ζ(2p+1) for any positive value of p. The integrand consists of the product of a polynomial by one or two elementary trigonometric functions. Examples for the first values of the argument are given. Some of them were already derived by other methods.

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