Congruences among generalized Bernoulli numbers

Abstract

The number B0,χ equals φ(M)/M (φ is Euler’s phi-function) if χ is the principal character and 0 otherwise. If m ≥ 1, then Bm,χ = 0 if χ(−1) = (−1)m−1 (unless M = m = 1). For m > 1 the converse is also true, by (2) and the functional equation of L(s, χ), but we will not use this. We are going to study some objects related to quadratic characters. Let d be the discriminant of a quadratic field, and denote by χ d = ( d · ) the associated quadratic character (Kronecker symbol). The numbers Bm,χ d /m are always integers unless d = −4 or d = ±p, where p is an odd prime number such that 2m/(p − 1) is an odd integer, in which case they have denominator 2 or p, respectively (cf. [3] or [6]). We also have the case d = 1 for which χ d is the trivial character; in this case, the denominator of Bm/m contains exactly those primes p for which p − 1 divides m. Together, these numbers d are the so-called fundamental discriminants (they can also be described as the set of square-free numbers of the form 4n+ 1 and 4 times square-free numbers not of this form) and the corresponding characters χ d give all primitive quadratic characters.