On the distribution of $\log |L(\sigma ,\chi )|$ and $\log L(\sigma ,{\chi }_D)$ in the modulus aspect

Abstract

Let $\chi$ be a primitive Dirichlet character whose conductor $q$ is a prime number. For the certain averages of values of $\log |L(s,\chi )|$ in $q$-aspect at a fixed $s=\sigma >1/2$, under Generalized Riemann Hypothesis (GRH), we explain it can be written as integrals involving the same density function ($M$-function) for the average of values of the difference between the logarithms of two symmetric power $L$-functions in the level aspect. For the distribution of values of $\log L(s,{\chi }_D)$ and $L^{\prime }/L(s,{\chi }_D)$ in the $D$-aspect at a fixed $s=\sigma >1/2$ which $L({\sigma }^{\prime },\chi )\ne 0$ in $\sigma \le {\sigma }^{\prime }\le 1$, where ${\chi }_D$ is a real character attached to a fundamental discriminant $D$, we construct a $M$-function unconditionally.