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Zbl 0738.11023
Girstmair, Kurt
A theorem on the numerators of the Bernoulli numbers. (English)
[J] Am. Math. Mon. 97, No.2, 136-138 (1990). ISSN 0002-9890

Classically, the Bernoulli numbers $B\sb m$ are defined by $t/(e\sp t- 1)=\sum\sp \infty\sb{m=0}B\sb mt\sp m/m!$. These numbers are rational and, for odd $m\ge 3$, $B\sb m=0$. For even $m\ge0$, $B\sb m\ne 0$ and we can write uniquely $B\sb m=N\sb m/D\sb m$, where $N\sb m, D\sb m$ are relatively prime integers and $D\sb m\ge 1$. The following theorem concerning the numerators $N\sb m$ is due to von Staudt (1845): ``Let $m\ge2$ be even, and $p$ a prime with $(p-1)\dag m$. If $p\sp r$ divides $m$ ($r\ge1$), then $p\sp r$ divides $N\sb m$, too.''\par 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 $B\sb{m,k}$ $(0\le k\le n-1)$ defined as follows $$t/(\zeta\sp k\cdot e\sp t-1)=\sum\sp \infty\sb{m=0}B\sb{m,k}t\sp m/m!,$$ where $\zeta=e\sp{2\pi i/n}$ is a primitive $n$th root of unity for $n\ge 2$.
[L.Skula (Brno)]

MSC 2000:: *11B68 Bernoulli numbers, etc.
11S80 Other analytic theory of local fields

Keywords: cyclotomic Bernoulli numbers; von Staudt theorem; Bernoulli numbers; numerators

Cited in: Zbl 0738.11024

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