Congruences concerning Bernoulli numbers and Bernoulli polynomials
Let \(B_n(x)\), resp. \(B_n\), denote the classical Bernoulli polynomial, resp. number. In the paper under review the author proves some generalizations of Kummer's congruence by determining \[ \frac{B_{k(p-1)+b}(x)}{(k(p-1)+b)}\pmod{p^n} \] where \(p\) is an odd prime, \(x\) a \(p\)-integral rational number and \(p-1\nmid b\), while Kummer considered only the case \(n=1\). As an application he proves explicit formulae for \[ \sum_{x=1}^{p-1}\frac 1{x^k}\pmod{p^3}, \] and, for example, \[ (p-1)!\equiv \frac{pB_{2p-2}}{2p-2}-\frac{pB_{p-1}}{p-1}- \frac 12 \left(\frac{pB_{p-1}}{p-1}\right)^2\pmod{p^3},\;p>3. \] In a similar way he considers also generalized Bernoulli numbers \(B_{n,\chi}\), where \(\chi\) denotes a Dirichlet character.
scientific article; zbMATH DE number 1512456
scientific article