Fast converging series for zeta numbers in terms of polynomial representations of Bernoulli numbers

TL;DR

This paper introduces a new polynomial representation of Bernoulli numbers that enhances understanding and enables efficient computation of zeta function values like ζ(3), ζ(5), and ζ(7).

Contribution

It presents a novel polynomial-based method for Bernoulli numbers that simplifies the calculation of zeta function values and their derivatives.

Findings

Polynomial representation of Bernoulli numbers derived.

Efficient computation of ζ(3), ζ(5), ζ(7) demonstrated.

Provides a new approach for calculating zeta function derivatives.

Abstract

In this work we introduce a new polynomial representation of the Bernoulli numbers in terms of polynomial sums allowing on the one hand a more detailed understanding of their mathematical structure and on the other hand provides a computation of $B_{2n}$ as a function of B$_{2n-2}$ only. Furthermore, we show that a direct computation of the Riemann zeta-function and their derivatives at k $\in \mathbb Z$ is possible in terms of these polynomial representation. As an explicit example, our polynomial Bernoulli number representation is applied to fast approximate computations of $\zeta$(3), $\zeta$(5) and $\zeta$(7).