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

Abstract

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 $\left|{\zeta }^{\prime }\left({\sigma }_A+it\right)/\zeta \left({\sigma }_A+it\right)\right|⩾\left(\left(e^A-1\right)/A\right){\log }_{2}T+O\left({\log }_{2}T/{\log }_{3}T\right)$ has a solution $t\in [T^{\beta },T]$ for all sufficiently large $T,$ where ${\sigma }_A=1-A/{\log }_{2}T.$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.