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

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).