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

## TL;DR

This paper explores the connection between the maximum of the Riemann zeta function on the 1-line and the maximal order of the error term in zero counting, linking conjectures and hypotheses in number theory.

## Contribution

It demonstrates that conjectured bounds on the error term and the Riemann hypothesis imply Littlewood's conjecture on the zeta function's maximum.

## Key findings

- Conjectured bounds on S(t) imply the asymptotic behavior of |ζ(1+it)|.
- The Riemann hypothesis combined with bounds on S(t) supports Littlewood's conjecture.
- The relationship between the zeta function's maximum and error terms is extended to 1/2<σ<1 region.

## 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.