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

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 $$\beta=2+\frac{\log(1-6/\pi^2)}{\log(2)}\approx 0.6491\ldots$$.

### MSC:

 11B68 Bernoulli and Euler numbers and polynomials

### Keywords:

Bernoulli numbers
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### Digital Library of Mathematical Functions:

§24.9 Inequalities ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials