Notes on zeta ratio stabilization

TL;DR

This paper explores how recent homological stability ideas can address the Ratios Conjecture over function fields, achieving uniform power savings and implications for zero statistics.

Contribution

It demonstrates the application of homological stability techniques to the Ratios Conjecture over $\

Findings

Achieves uniform power saving at certain distances from the critical line.

Shows cancellation beyond GRH in large ranges of moduli.

Provides insights into low-lying zeros statistics.

Abstract

This semi-expository note clarifies the extent to which recent ideas in homological stability can resolve the Ratios Conjecture over $\mathbb{F}_q(t)$. For large fixed $q$, a uniform power saving at distance $\ge q^{-\delta}$ from the critical line is possible. This implies cancellation-beyond-GRH in arbitrarily large ranges of moduli relative to the family of $L$-functions. It has applications to the statistics of low-lying zeros.