New series representations for zeta numbers using polylogarithmic identities in combination with a polynomial description of Bernoulli numbers

TL;DR

This paper introduces a new series representation for b6(3) using polylogarithmic identities and Bernoulli numbers, achieving faster convergence and high-precision numerical results compared to existing methods.

Contribution

The authors develop a novel series for b6(3) that converges faster than previous Clausen-based series by combining polylogarithmic identities with polynomial Bernoulli number descriptions.

Findings

Achieved b6(3) approximation with 2^{-26} accuracy using only four series terms.

Demonstrated the new series surpasses BBP-type formulas in convergence speed.

Provided explicit numerical comparison with existing BBP formulas.

Abstract

With this paper we introduce a new series representation of $\zeta(3)$, which is based on the Clausen representation of odd integer zeta values. Although, relatively fast converging series based on the Clausen representation exist for $\zeta(3)$, their convergence behavior is very slow compared to BBP-type formulas, and as a consequence they are not used for explicit numerical computations. The reason is found in the fact that the corresponding Clausen function can be calculated analytically for a few rational arguments only, where $x=\frac{1}{6}$ is the smallest one. Using polylogarithmic identities in combination with a polynomial description of the even Bernoulli numbers, the convergence behavior of the Clausen-type representation has been improved to a level that allows us to challenge ultimately all BBP-type formulas available for $\zeta(3)$. We present an explicit numerical comparison between one of the best available BBP formulas and our formalism. Furthermore, we demonstrate by an explicit computation using the first four terms in our series representation only that $\zeta(3)$ results with an accuracy of $2*10^{-26}$, where our computation guarantees on each approximation level for an analytical expression for $\zeta(3)$.