# Is the Riemann zeta function in a short interval a 1-RSB spin glass ?

## TL;DR

This paper explores the connection between the Riemann zeta function's local maxima and 1-RSB spin glasses, showing that the maxima are strongly clustered, which supports a spin glass analogy.

## Contribution

It introduces a random model of the Riemann zeta function and demonstrates that its two-overlap distribution matches that of a 1-RSB spin glass, advancing the understanding of its complex structure.

## Key findings

- Two-overlap distribution matches 1-RSB spin glass
- Local maxima are strongly clustered
- Supports spin glass analogy for zeta maxima

## Abstract

Fyodorov, Hiary & Keating established an intriguing connection between the maxima of log-correlated processes and the ones of the Riemann zeta function on a short interval of the critical line. In particular, they suggest that the analogue of the free energy of the Riemann zeta function is identical to the one of the Random Energy Model in spin glasses. In this paper, the connection between spin glasses and the Riemann zeta function is explored further. We study a random model of the Riemann zeta function and show that its two-overlap distribution corresponds to the one of a one-step replica symmetry breaking (1-RSB) spin glass. This provides evidence that the local maxima of the zeta function are strongly clustered.