Connection between the arithmetic properties of generalized Bernoulli numbers (Q1803500)

In this paper the Bernoulli numbers \(B^ m_ f\) belonging to \(f\) are defined, where \(f\) is a periodic mapping from \(\mathbb{Z}\) into \(\mathbb{C}\) with period \(n\) \((n\) positive integer, \(m\) nonnegative integer), as follows: \[ \sum^ n_{k=1}f(k)e^{kt}t/(e^{nt}-1)=\sum^ \infty_{m=0}B^ m_ ft^ m/m!\;. \] The two cases are considered: 1) \(f\) is an additive character mod \(n\), 2) \(f\) is a primitive Dirichlet character mod \(n\). In the first case we get the numbers \(B^ m_ n\) \((f(k)=e^{2\pi ik/n})\) introduced by the author [Monatsh. Math. 104, 109-118 (1987; Zbl 0626.12001)] and in the second case the generalized Bernoulli numbers \(B^ m_ \chi\) \((f=\chi)\) introduced by \textit{H. W. Leopoldt} [Abh. Math. Semin. Univ. Hamburg 22, 131-140 (1958; Zbl 0080.030)]. Both types of numbers have a great significance in the area of algebraic number fields. These numbers satisfy the classical theorems for the ordinary Bernoulli numbers: von Staudt-Clausen theorem and the Kummer congruence. (The Kummer congruence for the Leopoldt generalized Bernoulli numbers \(B^ m_ \chi\) was shown by \textit{L. Carlitz} [J. Reine Angew. Math. 202, 174- 182 (1959; Zbl 0125.022)]). To show the similarity of these arithmetical properties of the considered Bernoulli numbers Leopoldt's character coordinate \(y_ \chi\): \(\mathbb{Q}(e^{2\pi in})\to\mathbb{Q}(\chi,k)\) \((k\in\mathbb{Z})\) is investigated in detail. This mapping maps \(B^ m_ n\) onto \(B^ m_ \chi\).