An Alternative Form of the Functional Equation for Riemann's Zeta Function, II

TL;DR

This paper presents a new, simple proof of the symmetric functional equation for Riemann's Zeta function and related functions, introduces a new entire function connected to these equations, and explores implications for zeros of key functions.

Contribution

It provides a novel, direct proof of the symmetric functional equation and introduces a new entire function related to Riemann's Zeta function, extending understanding of their zeros.

Findings

New proof of the symmetric functional equation for Riemann's Zeta and related functions

Introduction of a new entire function { ext{ extcyr ext{E}}}(s) with analytic continuation

Extended results on the zeros of Riemann Zeta and Dirichlet Beta functions

Abstract

This paper treats about one of the most remarkable achievements by Riemann, that is the symmetric form of the functional equation for {\zeta}(s). We present here, after showing the first proof of Riemann, a new, simple and direct proof of the symmetric form of the functional equation for both the Eulerian Zeta function and the alternating Zeta function, connected with odd numbers. A proof that Euler himself could have arranged with a little step at the end of his paper "Remarques sur un beau rapport entre les s\'eries des puissances tant direct que r\'eciproches". This more general functional equation gives origin to a special function, here named {\cyr \E}(s), which we prove that it can be continued analytically to an entire function over the whole complex plane using techniques similar to those of the second proof of Riemann. Moreover we are able to obtain a connection between Jacobi's imaginary transformation and an infinite series identity of Ramanujan. Finally, after studying the analytical properties of the function {\cyr \E}(s), we complete and extend the proof of a Fundamental Theorem, both on the zeros of Riemann Zeta function and on the zeros of Dirichlet Beta function, using also the Euler-Boole summation formula.