# Extreme values of the Riemann zeta function and its argument

## TL;DR

This paper establishes new lower bounds for the maximum size of the Riemann zeta function and its argument on large intervals, using a combination of resonance methods and convolution formulas, with results conditional on the Riemann hypothesis.

## Contribution

It introduces a novel approach combining resonance methods with convolution formulas to derive lower bounds for zeta function maxima and argument functions.

## Key findings

- Maximum of |z(1/2+it)| exceeds rac{1}{2} imes ext{exp}( ext{sqrt(log T log log log T / log log T)})
- Conditional lower bounds for the argument S(t) and its integral S_1(t) are established under the Riemann hypothesis.
- Results improve understanding of extreme values of the zeta function and its argument on large intervals.

## Abstract

We combine our version of the resonance method with certain convolution formulas for $\zeta(s)$ and $\log\, \zeta(s)$. This leads to a new $\Omega$ result for $|\zeta(1/2+it)|$: The maximum of $|\zeta(1/2+it)|$ on the interval $1 \le t \le T$ is at least $\exp\left((1+o(1)) \sqrt{\log T \log\log\log T/\log\log T}\right)$. We also obtain conditional results for $S(t):=1/\pi$ times the argument of $\zeta(1/2+it)$ and $S_1(t):=\int_0^t S(\tau)d\tau$. On the Riemann hypothesis, the maximum of $|S(t)|$ is at least $c \sqrt{\log T \log\log\log T/\log\log T}$ and the maximum of $S_1(t)$ is at least $c_1 \sqrt{\log T \log\log\log T/(\log\log T)^3}$ on the interval $T^{\beta} \le t \le T$ whenever $0\le \beta < 1$.