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

TL;DR

This paper proves an omega theorem for the logarithmic derivative of the Riemann zeta function near the 1-line, showing it attains large values in certain intervals, with implications for Dirichlet L-functions.

Contribution

It establishes a new omega theorem for the logarithmic derivative of zeta and L-functions near the 1-line using the resonance method, including conditional measure bounds.

Findings

Logarithmic derivative of zeta exceeds bounds near the 1-line

Existence of solutions for large values in specified intervals

Generalization to Dirichlet L-functions

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 $\left| \zeta^{\prime}\left(\sigma_A+it\right)/\zeta\left(\sigma_A+it\right) \right| \geqslant \left(\left(e^A-1\right)/A\right)\log_2 T + O\left(\log_2 T / \log_3 T\right)$ 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.