## Arithmetic properties of generalized Bernoulli numbers.(English)Zbl 0125.02202

Let $$f$$ be a fixed integer $$\geq 1$$ and $$\chi(r)$$ a primitive character $$\text{mod}\,f$$. H.-W. Leopoldt [Abh. Math. Semin. Univ. Hamb. 22, 131–140 (1958; Zbl 0080.03002)] has defined numbers $$B_\chi{}^n,B_\chi{}^n(x)$$ by means of $\sum_{r=1}^f\chi(r)\frac{te^{(r+x)t}}{e^{ft}-1}=\sum_{n=1}^\infty B_\chi{}^n(x)\frac{t^n}{n!},\quad B_\chi{}^n=B_\chi{}^n(0).$ For $$f=1$$, $$\chi$$ is the principal character, and $$B_\chi{}^n$$ reduces to the ordinary Bernoulli numbers; for $$f=4$$ and $$\chi$$ the non-principal character $$(\text{mod}\,4)$$, $$B_\chi{}^n$$ reduces to $$-{\textstyle\frac 1{2}}(n+1)E_{n+1}$$, where $$E_{n+1}$$ is an Euler number. In the present paper the author first extends one of Leopoldt’s results by showing that a theorem of the Staudt-Clausen type holds for the number $$n^{-1}B_\chi{}^n$$; he also establishes the related result that if $$p$$ is a rational prime such that $$p^e|n$$ but $$p\nmid f$$, then $$p^e$$ divides the numerator of $$B_\chi{}^n$$. Next he derives congruences of the Kummer type for $$B_\chi{}^n$$. Finally he proves that if $$p^{e-1}(p-1)|m$$ but $$p\nmid f$$, then $\frac 1{m+1}B_\chi{}^{m+1}\equiv\frac 1{f}(1-\chi(p))\sum_{r=1}^fr\chi(r)\ (\text{mod}\,p^e). (*)$ In particular for $$f=4$$, $$(*)$$ reduces to a known result for Euler numbers.

### MSC:

 11B68 Bernoulli and Euler numbers and polynomials

Zbl 0080.03002
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