# Extreme values of the Riemann zeta function on the 1-line

## TL;DR

This paper establishes new lower bounds for the extreme values of the Riemann zeta function on the 1-line, matching conjectured predictions, by developing a novel variant of the long resonator method that leverages self-similarity properties.

## Contribution

Introduces a new variant of the long resonator method using self-similarity, improving the analysis of zeta function maxima without sparsification.

## Key findings

- Proves arbitrarily large values of |(1+it)| with explicit lower bounds.
- Matches the conjectured optimal lower bounds for zeta function maxima.
- Develops a new resonator technique exploiting self-similarity properties.

## Abstract

We prove that there are arbitrarily large values of $t$ such that $|\zeta(1+it)| \geq e^{\gamma} (\log_2 t + \log_3 t) + \mathcal{O}(1)$. This essentially matches the prediction for the optimal lower bound in a conjecture of Granville and Soundararajan. Our proof uses a new variant of the "long resonator" method. While earlier implementations of this method crucially relied on a "sparsification" technique to control the mean-square of the resonator function, in the present paper we exploit certain self-similarity properties of a specially designed resonator function.