The mean value of $|L(k,\chi)|^2$ at positive rational integers $k\ge 1$
Volume 90 / 2001
Colloquium Mathematicum 90 (2001), 69-76
MSC: Primary 11M06, 11M20, 11R18.
DOI: 10.4064/cm90-1-6
Abstract
Let $k\ge 1$ denote any positive rational integer. We give formulae for the sums $$ S_{\rm odd}(k,f) =\sum _{\chi (-1)=-1}| L(k,\chi )| ^2 $$ (where $\chi $ ranges over the $\phi (f)/2$ odd Dirichlet characters modulo $f>2$) whenever $k\ge 1$ is odd, and for the sums $$ S_{\rm even}(k,f) =\sum _{\chi (-1)=+1} | L(k,\chi )| ^2 $$ (where $\chi $ ranges over the $\phi (f)/2$ even Dirichlet characters modulo $f>2$) whenever $k\ge 1$ is even.
