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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/(e t -1)= m=0 B m t m /m!. These numbers are rational and, for odd m3, B m =0. For even m0, B m 0 and we can write uniquely B m =N m /D m , where N m ,D m are relatively prime integers and D m 1. The following theorem concerning the numerators N m is due to von Staudt (1845): “Let m2 be even, and p a prime with (p-1)m. If p r divides m (r1), 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 (0kn-1) defined as follows

t/(ζ k ·e t -1)= m=0 B m,k t m /m!,

where ζ=e 2πi/n is a primitive nth root of unity for n2.

Reviewer: L.Skula (Brno)

MSC:
11B68Bernoulli and Euler numbers and polynomials
11S80Other analytic theory of local fields