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Ein v. Staudt-Clausenscher Satz für periodische Bernoullizahlen

A theorem of von Staudt-Clausen type for periodic Bernoulli numbers

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Abstract

Letk, n be relatively prime integers, 1≤k<n. The numbersB m k =m(i/2) mcot(m−1)(πk/n),m≥2, belong to then-th cyclotomic field. They occur as values of certain simple Dirichlet series at integral arguments and have a natural interpretation as generalized Bernoulli numbers. Therefore the arithmetic properties of these numbers arouse some interest. Ifn is not a prime power,B m k /m is an algebraic integer. In the other case, i.e.n=p e,p prime, the fractional part ofB m k /m resp.B m k is determined in this paper. It is periodic modulop−1 and exhibits a connection between these Bernoulli numbers and the Stirling numbers of the second kind.

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  1. Institut für Mathematik, Universität Innsbruck, Technikerstraße 25, A-6020, Innsbruck, Österreich

    Kurt Girstmair

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  1. Kurt Girstmair
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Girstmair, K. Ein v. Staudt-Clausenscher Satz für periodische Bernoullizahlen. Monatshefte für Mathematik 104, 109–118 (1987). https://doi.org/10.1007/BF01326783

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  • DOI: https://doi.org/10.1007/BF01326783

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