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

J. Braun; D. Romberger; H. J. Bentz

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arXiv · 1609.03036

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

J. Braun·D. Romberger·H. J. Bentz

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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 $\zeta(3)$.

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BibTeX RIS

J. Braun, D. Romberger, H. J. Bentz. 2016-09-10. New series representations for zeta numbers using polylogarithmic identities in combination with a polynomial description of Bernoulli numbers. https://arxiv.org/abs/1609.03036

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