The Hurwitz Zeta Function at the Positive Integers

2019/02/19 by Jose Risomar Sousa, Sousa, Jose Risomar
Mathematics · Physics and Astronomy · #11-XX #Advanced Mathematical Identities #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT) #Quantum Mechanics and Applications

paper · pdf · doi:10.48550/arxiv.1902.06885

openalex publication_date 2019/02/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A formula for the Hurwitz zeta function at the positive integers k, ζ(k,b), is created by solving the real and the imaginary parts separately and then combining them. A few different formulae for the Hurwitz zeta function are known from the literature, but they are very general and usually hold for \Re(k)>1. The advantage of formulae that only hold at the positive integers is the fact that they are simpler and easier to work with. An analytic continuation of the generating function of ζ(k,b) is also obtained as ∑_{k≥ 2}x^k(ζ(k,b)-1/b^k), where the term 1/b^k was subtracted for convenience.

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