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

Classically, the Bernoulli numbers ${B}_{m}$ are defined by $t/\left({e}^{t}-1\right)={\sum }_{m=0}^{\infty }{B}_{m}{t}^{m}/m!$. These numbers are rational and, for odd $m\ge 3$, ${B}_{m}=0$. For even $m\ge 0$, ${B}_{m}\ne 0$ and we can write uniquely ${B}_{m}={N}_{m}/{D}_{m}$, where ${N}_{m},{D}_{m}$ are relatively prime integers and ${D}_{m}\ge 1$. The following theorem concerning the numerators ${N}_{m}$ is due to von Staudt (1845): “Let $m\ge 2$ be even, and $p$ a prime with $\left(p-1\right)†m$. If ${p}^{r}$ divides $m$ ($r\ge 1$), 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 ${B}_{m,k}$ $\left(0\le k\le n-1\right)$ defined as follows

$t/\left({\zeta }^{k}·{e}^{t}-1\right)=\sum _{m=0}^{\infty }{B}_{m,k}{t}^{m}/m!,$

where $\zeta ={e}^{2\pi i/n}$ is a primitive $n$th root of unity for $n\ge 2$.

Reviewer: L.Skula (Brno)

##### MSC:
 11B68 Bernoulli and Euler numbers and polynomials 11S80 Other analytic theory of local fields