Instant Evaluation and Demystification of $\zeta(n),L(n,\chi)$ that Euler,Ramanujan Missed - I

TL;DR

This paper introduces a method for instant evaluation of the Riemann Zeta function and Dirichlet L series at positive integers using Taylor series expansions and functional equations, simplifying calculations and providing new finite sum expressions.

Contribution

It presents a novel approach to evaluate special functions at positive integers instantly, including finite sum formulas, by analyzing Taylor series and functional equations.

Findings

Instant evaluation formulas for Riemann Zeta at positive even integers.

Finite sum expressions for Dirichlet L series at argument one.

Extension of methods to Lerch's Zeta function.

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.