On expansions involving the Riemann zeta function and its derivatives

TL;DR

This paper explores spectral properties of the fractional part function to derive rational approximations of the Riemann zeta function and its derivatives across certain vertical lines in the complex plane.

Contribution

It introduces a novel spectral approach to approximate the Riemann zeta function and derivatives, expanding understanding of their behavior in the right half-plane.

Findings

Rational approximations valid for Re s > 1/2 and Re s > 0

Spectral analysis of the fractional part function

Explicit computations related to the fractional part function

Abstract

By studying the spectral aspects of the fractional part function in a well-known separable Hilbert space, we show, among other things, a rational approximation of the Riemann zeta function and its derivatives valid on every vertical line in the right half-planes $\Re s> 1/2$ and $\Re s >0.$ Moreover, we provide some discussions and explicit computations related to the fractional part function.