On Large Values of $|\zeta(\sigma+{\rm i}t)|$

TL;DR

This paper studies the extreme values of the Riemann zeta function, providing improved bounds on its maximum magnitude on the 1-line and within the half-critical strip, advancing understanding of its growth behavior.

Contribution

It offers new lower bounds for the maximum of |z(s)| on the 1-line and in the half-critical strip, improving previous results through refined techniques and bounds.

Findings

Established a lower bound for |z(1+it)| involving ^b3 and iterated logarithms.

Derived an improved lower bound for (s) in the half-critical strip as (s) rac{(\u007flog T)^{1-s}}{(\u007flog_2 T)^s}.

Enhanced previous bounds by leveraging improved GCD sum estimates.

Abstract

We investigate the extreme values of the Riemann zeta function $\zeta(s)$. On the 1-line, we obtain a lower bound evaluation $$\max_{t\in[1,T]}|\zeta(1+\i t)|\ge {\rm e}^\gamma(\log_2T+\log_3T+c),$$ with an effective constant $c$ which improves the result of Aistleitner, Mahatab and Munsch. In the half-critical strip $1/2<\re s<1$, we get an improved $c(\sigma)$ in the evaluation $$\max_{t\in[0,T]}\log|\zeta(\sigma+\i t)|\ge c(\sigma)\frac{(\log T)^{1-\sigma}}{(\log_2T)^\sigma},$$ when $\sigma\searrow 1/2$, based on an improved lower bound of GCD sums. This improves the result of Bondarenko and Seip.