A new expansion of the Riemann zeta function

TL;DR

This paper introduces a novel convergent series expansion of the Riemann zeta function within the critical strip, utilizing polynomials with roots on the critical line and coefficients linked to fundamental constants.

Contribution

It presents a new expansion of the Riemann zeta function as a convergent series with polynomials having roots on the critical line, connecting coefficients to key mathematical constants.

Findings

Series converges within the critical strip

Polynomials have roots on the line Re(s)=1/2

Coefficients involve Euler's constant and zeta values

Abstract

After a brief introduction to Ramanujan's method of summation, we give an expansion of the Riemann Zeta function in the critical strip as a convergent series $\sum_{m\geq 0}x_m P_m(s) $ where the functions $P_m$ are polynomials with their roots on the line $\{\Re(s)=1/2\}$, the coefficients $x_m$ being finite linear combinations of the Euler constant $\gamma$ and the values $\zeta(2),\zeta(3),\dots,\zeta(m+1).$