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

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

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arXiv · 1503.04636

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

J. Braun·D. Romberger·H. J. Bentz

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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).

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BibTeX RIS

J. Braun, D. Romberger, H. J. Bentz. 2015-03-16. Fast converging series for zeta numbers in terms of polynomial representations of Bernoulli numbers. https://arxiv.org/abs/1503.04636

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