A note on the maximum of the Riemann zeta function on the 1-line

Abstract

We investigate the relationship between the maximum of the zeta function on the 1-line and the maximal order of $S(t)$, the error term in the number of zeros up to height $t$. We show that the conjectured upper bounds on $S(t)$ along with the Riemann hypothesis imply a conjecture of Littlewood that $\max_{t\in [1,T]}|\zeta(1+it)|\sim e^\gamma\log\log T$. The relationship in the region $1/2<\sigma<1$ is also investigated.