## A theorem on the numerators of the Bernoulli numbers.(English)Zbl 0738.11023

Classically, the Bernoulli numbers $$B_ m$$ are defined by $$t/(e^ t- 1)=\sum^ \infty_{m=0}B_ mt^ m/m!$$. These numbers are rational and, for odd $$m\geq 3$$, $$B_ m=0$$. For even $$m\geq0$$, $$B_ m\neq 0$$ and we can write uniquely $$B_ m=N_ m/D_ m$$, where $$N_ m, D_ m$$ are relatively prime integers and $$D_ m\geq 1$$. The following theorem concerning the numerators $$N_ m$$ is due to von Staudt (1845): “Let $$m\geq2$$ be even, and $$p$$ a prime with $$(p-1)† m$$. If $$p^ r$$ divides $$m$$ ($$r\geq1$$), then $$p^ r$$ divides $$N_ m$$, too.”
A great number of mathematicians have given various proofs of this theorem since. The author presents quite a new proof based on the notion of “cyclotomic” Bernoulli numbers $$B_{m,k}$$ $$(0\leq k\leq n-1)$$ defined as follows $t/(\zeta^ k\cdot e^ t-1)=\sum^ \infty_{m=0}B_{m,k}t^ m/m!,$ where $$\zeta=e^{2\pi i/n}$$ is a primitive $$n$$th root of unity for $$n\geq 2$$.
Reviewer: L.Skula (Brno)

### MSC:

 11B68 Bernoulli and Euler numbers and polynomials 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.)
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