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

2015/03/16 by Jürgen Braun, Braun, J., D. Romberger +3
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1503.04636

openalex publication_date 2015/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ∈ \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 ζ(3), ζ(5) and ζ(7).

Related

• New series representations for zeta numbers using polylogarithmic identities in combination with a polynomial description of Bernoulli numbers
• A New Representation of the Riemann Zeta Function ζ(s)
• A new understanding of ζ(k)
• A new formula for ζ(s)
• New rapidly converging series representations for values of the Riemann zeta function and the Dirichlet beta function