Remainder Pad\'e approximants for the Hurwitz zeta function

TL;DR

This paper develops Remainder Padé approximants for the Hurwitz zeta function, proves their convergence on the positive real line, and applies them to obtain new rational approximations of erzi and erzi.

Contribution

It introduces a novel application of Remainder Pade9 approximants to the Hurwitz zeta function and proves their convergence, enabling new rational approximations of specific zeta values.

Findings

Proved convergence of Remainder Pade9 approximants on the positive real line.

Constructed new rational approximations of erzi(2) and erzi(3).

Demonstrated the effectiveness of the method for special functions.

Abstract

Following our earlier research, we use the method introduced by the author in \cite{prevost1996} named Remainder Pad\'e Approximant in \cite{rivoalprevost}, to construct approximations of the Hurwitz zeta function. We prove that these approximations are convergent on the positive real line. Applications to new rational approximations of $\zeta(2)$ and $\zeta(3)$ are given.