Character average of fourth power of Dirichlet L-series at unity

Abstract

For a Dirichlet character modulo an integer $q\ge 3$, we use a highly simple elementary method to give an asymptotic formula for $\sum _{\chi \ne \chi _0 ({\bmod q})} {|L(1, \chi )|^{4}} ,$ where $\chi _0 ({\bmod \, q})$ is the principal character. This result seems to be new.