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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,ψ play an important role in many fields of number theory. Here ψ denotes a character with the conductor g, and the numbers B n,ψ are defined by

a=1 g ψ(a)te at e gt -1= n=0 B n,ψ t n n!·

Let p>3 be a prime, θ be a character with conductor q and θ(p)=1, χ a character which differs from θ only by the Legendre symbol modulo p then the following congruences (Theorem 1) hold: B m+1,χ 0(modp l ), where l 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,χ B r,θ (modp 2l ) if χ(-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.

MSC:
11B68Bernoulli and Euler numbers and polynomials
11R11Quadratic extensions
11R29Class numbers, class groups, discriminants
11S40Zeta functions and L-functions of local number fields