Omega Theorems for Logarithmic Derivatives of Zeta and L-functions Near the 1-line

We establish an omega theorem for logarithmic derivative of the Riemann zeta function near the 1-line by resonance method. We show that the inequality $\mid {\zeta }^{\mathrm{\prime }}({\sigma }_A+it)\mathrm{/}\zeta ({\sigma }_A+it)\mid ⩾\left((e^A-1)\mathrm{/}A\right){\log }_{2}T+O\left({\log }_{2}T\mathrm{/}{\log }_{3}T\right)$ has a solution $t\in [T^{\beta },T]$ for all sufficiently large $T,$ where ${\sigma }_A=1-A\mathrm{/}{\log }_{2}T\mathrm{.}$Furthermore, we give a conditional lower bound for the measure of the set of $t$ for which the logarithmic derivative of the Riemann zeta function is large. Moreover, similar results can be generalized to Dirichlet $L$-functions.