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

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.