On modification of the {$q$}-{$L$}-series and its applications

Abstract

We slightly modify the definitions of $q$-Hurwitz $\zeta$-functions and $q$-$L$-series constructed by J. Satoh. By using these modified functions, we give some relations for the ordinary Dirichlet $L$-series. Especially we give an elementary proof of Katsurada's formula on the values of Dirichlet $L$-series at positive integers.