Instant Evaluation and Demystification of ζ(n),L(n,χ) that Euler,Ramanujan Missed - I

2008/01/06 by Vivek V. Rane, Rane, Vivek V.
Mathematics · #11M35 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.0801.0884

openalex publication_date 2008/01/06 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28

Abstract

For Hurwitz Zeta function,we consider its Taylor series expansion about various points as an analytic function of second variable in appropriate discs.We show that these Taylor are all polynomials in second variable for a non positive integral argument in first variable.On using functionalequations this results in instant evaluation of Riemann Zeta function at positive even integral values of its argument and of Dirichlet L series at positive integral values of its argument,when the argument and the corresponding Dirichlet character are both even or both odd.We also obtain finite sum expression for any Dirichlet L series,when its argument is one.We also deal with Lerch's Zeta function on similar lines.

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