On the generalized Bernoulli numbers that belong to unequal characters

Abstract

The study of class number invariants of absolute abelian fields, the investigation of congruences for special values of $L$-functions, Fourier coefficients of half-integral weight modular forms, Rubin's congruences involving the special values of $L$-functions of elliptic curves with complex multiplication, and many other problems require congruence properties of the generalized Bernoulli numbers (see [16-18, 12, 29, 3], etc.). The first steps in this direction can be found in the papers of H. W. Leopoldt (see [15]) and L. Carlitz (see [5]). For further studies, see [22, 24, 29]. This paper presents some new examples extending both old author's results and recent investigations of H. Lang (see [14]), A. Balog, H. Darmon, K. Ono (see [3]), etc. On the whole the proved results are consequence of congruences connecting the generalized Bernoulli numbers that belong to unequal characters.