Notes on zeta ratio stabilization

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.