An application of the Mellin-Barnes type of integral to the mean square of \(L\)-functions
The mean square of all Dirichlet \(L\)-functions \((\text{mod }q)\) in the form \[ \varphi(q)^{-1} \sum_{\chi\pmod q}| L(s,\chi)|^2 \] are investigated. As usual \(\varphi(q)\) denotes Euler's function. A full asymptotic expansion is given which is a slight improvement of an earlier result of the author. The proof is based on a special case of the Mellin-Barnes integral expression for the Gauss hypergeometric function, namely \[ \Gamma(\alpha) (1-z)^{-\alpha}= \frac{1}{2\pi i} \int_{(b)} \Gamma(\alpha+s) \Gamma(-s) (-z)^s ds \] for \(| \arg(-z)| <\pi\) and \(-\text{Re }\alpha< b<0\). Here \((b)\) denotes the vertical straight line from \(b-i\infty\) to \(b+i\infty\).
mean square of Dirichlet \(L\)-functions
scientific article; zbMATH DE number 1262935
scientific article