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

Abstract

We investigate the extreme values of the Riemann zeta function $\zeta (s)$. On the 1-line, we obtain a lower bound evaluation $$\underset{t\in [1,T]}{\max }|\zeta (1+ıt)|\ge {\mathrm{e}}^{\gamma }({\log }_{2}T+{\log }_{3}T+c),$$ 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 $$\underset{t\in [0,T]}{\max }\log |\zeta (\sigma +ıt)|\ge c(\sigma )\frac{(\log T{)}^{1-\sigma }}{({\log }_{2}T{)}^{\sigma }},$$ when $\sigma \searrow 1/2$, based on an improved lower bound of GCD sums. This improves the result of Bondarenko and Seip.

19 pages