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

J. Braun, D. Romberger and H. J. Bentz
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 23, 2017, Number 2, Pages 54–80
Full paper (PDF, 139 Kb)

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Authors and affiliations

J. Braun
Department Chemie, Ludwig-Maximilians-Universität
München, 81377 München, Germany

D. Romberger
Fakultät IV, Hochschule Hannover
Ricklinger Stadtweg 120, 30459 Hannover, Germany

H. J. Bentz
Institut für Mathematik und Angewandte Informatik
Samelsonplatz 1, 31141 Hildesheim, Germany

Abstract

In this paper 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 B2n as a function of B2n − 2. Furthermore, we show that a direct computation of the Riemann zeta-function and their derivatives at k ∈ 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 ζ(3), ζ(5) and ζ(7).

Keywords

  • Bernoulli numbers
  • Bendersky’s L-numbers
  • Riemann zeta function

AMS Classification

  • 11B68

References

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Cite this paper

Braun, J., Romberger, D. & Bentz, H. J. (2017). Fast converging series for zeta numbers in terms of polynomial representations of Bernoulli numbers. Notes on Number Theory and Discrete Mathematics, 23(2), 54-80.

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