A Finite Reflection Formula For A Polynomial Approximation To The Riemann Zeta Function

TL;DR

This paper introduces a finite-sum polynomial approximation to the Riemann zeta function, deriving a reflection formula and analyzing its properties, including roots, residues, and connections to Bernoulli numbers.

Contribution

It presents a novel finite-sum approximation to ta(s) with a reflection formula and analyzes its mathematical properties and implications.

Findings

The approximation ta_w(N;s) has roots at s=-1 and s=0.

The residue at s=0 changes sign between N=176 and N=177.

Values at integer points relate to Bernoulli numbers.

Abstract

The Riemann zeta function can be written as the Mellin transform of the unit interval map w(x) = floor(1/x)*(-1+x*floor(1/x)+x) multiplied by s((s+1)/(s-1)). A finite-sum approximation to \zeta (s) denoted by \zeta_w(N;s) which has real roots at s=-1 and s=0 is examined and an associated function \chi (N ; s) is found which solves the reflection formula \zeta_w (N ; 1 - s) = \chi (N ; s) \zeta_w (N ; s). A closed-form expression for the integral of \zeta_w (N ; s) over the interval s=-1..0 is given. The function \chi (N ; s) is singular at s=0 and the residue at this point changes sign from negative to positive between the values of N=176 and N=177. Some rather elegant graphs of \zeta_w(N ; s) and the reflection functions \chi (N ; s) are also provided. The values \zeta_w (N ; 1 - n) for integer values of n are found to be related to the Bernoulli numbers.