The Euler numbers and recursive properties of Dirichlet L-functions

Abstract

Abstract The aim of this paper is using an elementary method and the properties of the Bernoulli polynomials to establish a close relationship between the Euler numbers of the second kind En∗ $E_{n}^{*}$ and the Dirichlet L-function L(s,χ) $L(s,\chi )$. At the same time, we also prove a new congruence for the Euler numbers En $E_{n}$. That is, for any prime p≡1mod8 $p\equiv 1\bmod 8$, we have Ep−32≡0modp $E_{\frac{p-3}{2}}\equiv 0\bmod p$. As an application of our result, we give a new recursive formula for one kind of Dirichlet L-functions.

Keywords

• Euler numbers
• Euler numbers of the second kind
• Bernoulli polynomials
• Dirichlet L-function
• Identity
• Congruence