Notes on zeta ratio stabilization

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.