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On the generalized Bernoulli numbers that belong to unequal characters. (English) Zbl 0965.11006
Congruence properties of generalized Bernoulli numbers $B_{n,\psi}$ play an important role in many fields of number theory. Here $\psi$ denotes a character with the conductor $g$, and the numbers $B_{n,\psi}$ are defined by $$ \sum_{a=1}^g \psi(a)t\frac{e^{at}}{e^{gt}-1}=\sum_{n=0}^\infty B_{n,\psi}\frac{t^n}{n!} . $$ Let $p>3$ be a prime, $\theta$ be a character with conductor $q$ and $\theta(p)=1$, $\chi$ a character which differs from $\theta$ only by the Legendre symbol modulo $p$ then the following congruences (Theorem 1) hold: $ B_{m+1,\chi}\equiv 0\pmod{p^l}$, where $l\in\Bbb N$ and $m=(p-1)p^{l-1}/2$. Continuing his studies the author proves some von Staudt’s type congruences of the form $B_{n,\chi}\equiv B_{r,\theta}\pmod{p^{2l}}$ if $\chi(-1)=(-1)^n$ and $r=sp^{3l-1}+n$, see Theorem 2. Finally he applies his results to quadratic number fields, showing that these congruences are $p$-adic approximations of the class number formula.
11B68Bernoulli and Euler numbers and polynomials
11R11Quadratic extensions
11R29Class numbers, class groups, discriminants
11S40Zeta functions and $L$-functions of local number fields
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