Conditional estimates for the logarithmic derivative of Dirichlet L-functions

Abstract

Assuming the Generalized Riemann Hypothesis, we establish explicit bounds in the $q$-aspect for the logarithmic derivative $\left(L^{\prime }/L\right)\left(\sigma ,\chi \right)$ of Dirichlet $L$-functions, where $\chi$ is a primitive character modulo $q\ge 10^{30}$ and $1/2+1/loglogq\le \sigma \le 1-1/loglogq$. In addition, for $\sigma =1$ we improve upon the result by Ihara, Murty and Shimura (2009). Similar results for the logarithmic derivative of the Riemann zeta-function are given.