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

Girstmair, Kurt

Am. Math. Mon. 97, No. 2, 136-138 (1990).

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.)

Keywords:

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