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

TL;DR

This paper introduces a new symmetric form of the functional equation for Riemann's Zeta function by deriving a special function A(s) related to Euler's earlier work, including integral representations and analytic continuation.

Contribution

It presents a novel symmetric form of the functional equation for Riemann's Zeta function through a new function A(s), connecting historical Euler equations with modern analysis.

Findings

Defined a new symmetric function A(s) related to Riemann's Zeta

Derived multiple integral representations of A(s)

Established an analytic continuation of A(s) using Ramanujan's identity

Abstract

In this paper we present a simple method for deriving an alternative form of the functional equation for Riemann's Zeta function. The connections between some functional equations obtained implicitly by Leonhard Euler in his work "Remarques sur un beau rapport entre les series des puissances tant directes que reciproques" in Memoires de l'Academie des Sciences de Berlin 17, (1768), permit to define a special function, named A(s), which is fully symmetric and is similar to Riemann's "XI" function. To be complete we find several integral representations of the A(s) function and as a direct consequence of the second integral representation we obtain also an analytic continuation of the same function using an identity of Ramanujan.