On the generalized Bernoulli numbers that belong to unequal characters (Q5929433)

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 NEWLINE\[NEWLINE \sum_{a=1}^g \psi(a)t\frac{e^{at}}{e^{gt}-1}=\sum_{n=0}^\infty B_{n,\psi}\frac{t^n}{n!} . NEWLINE\]NEWLINE 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 \(l\in\mathbb N\) and \(m=(p-1)p^{l-1}/2\). NEWLINENEWLINENEWLINEContinuing 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.