## Note on a congruence for p-adic L-functions.(English)Zbl 0565.12008

For a prime p, let $$\psi$$ be a Dirichlet character of p-power order. In this paper, G. Gras’ congruence [Sémin. Delange-Pisot-Poitou, 20e Année 1978/79, Théorie des nombres, Fasc. 2, Exp. 22 (1980; Zbl 0427.12014)] between the power series f(X,$$\chi)$$ and f(X,$$\chi$$ $$\psi)$$ representing p-adic L-functions is proved by using Iwasawa’s construction of these functions. An application of this congruence, the so-called Riemann-Hurwitz genus formula for the $$\lambda$$ $${}^-$$-invariants of imaginary abelian fields, is briefly discussed.

### MSC:

 11S40 Zeta functions and $$L$$-functions 11R18 Cyclotomic extensions

Zbl 0427.12014
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