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Notes on zeta ratio stabilization

Victor Y. Wang

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 $\ge q^{-δ}$ 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.

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 . For large fixed , a uniform power saving at distance from the critical line is possible. This implies cancellation-beyond-GRH in arbitrarily large ranges of moduli relative to the family of -functions. It has applications to the statistics of low-lying zeros.
Paper Structure (8 sections, 20 theorems, 112 equations)

This paper contains 8 sections, 20 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. General framework
  3. General analysis
  4. The quadratic case
  5. Proof of Theorem \ref{['THM:apply-stability-1']}
  6. Proof of Corollary \ref{['COR:integrate-stability-1']}
  7. Proof of Theorem \ref{['THM:one-level-density']}
  8. Discussion of geometric families

Key Result

Theorem 1.1

Fix integers $K,Q\ge 0$. Let $\delta = \max(576, 2016(K+Q))^{-1}$ as in EQN:quadratic-specialized-delta-value, and let $\omega = \frac{1}{84}$. Take an odd $q \ge 2^{12} 2^{1/\delta}$. Let $\tfrac{1}{2} - \delta < \operatorname{Re}(s_1),\dots,\operatorname{Re}(s_K) < \tfrac{1}{2} + \delta$ and $\ope where $\mathsf{RR}_L(\bm{s};n)$ is the main term in the Ratios Recipe of conrey2005integralconrey20

Theorems & Definitions (41)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 31 more