Sharp bounds for the Bernoulli numbers. (English) Zbl 0960.11016

Alzer, Horst

Arch. Math. 74, No. 3, 207-211 (2000).

The classical Bernoulli numbers \(B_n\) \((n=0,1,2\ldots)\) can be defined by \[ \frac{x}{e^x-1}=\sum_{n=0}^\infty B_n\frac{x^n}{n!}, \] \(|x|< 2\pi\). In the paper under review the author gives the best possible constants \(\alpha\) and \(\beta\) such that \[ \frac{2(2n)!}{(2\pi)^{2n}}\frac 1{1-2^{\alpha-2n}} \leq |B_{2n}|\leq \frac{2(2n)!}{(2\pi)^{2n}}\frac 1{1-2^{\beta-2n}} \] holds for all integers \(n\geq 1\), namely \(\alpha=0\) and \(\beta=2+\frac{\log(1-6/\pi^2)}{\log(2)}\approx 0.6491\ldots\).

Reviewer: Helmut Müller (Hamburg)

MSC:

11B68

Bernoulli and Euler numbers and polynomials

Keywords:

Bernoulli numbers