Conditional estimates for the logarithmic derivative of Dirichlet <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e20" altimg="si9.svg"><mml:mi>L</mml:mi></mml:math>-functions

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