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The positivity technique and low-lying zeros of Dirichlet $L$-functions

Tianyu Zhao

TL;DR

This paper develops and applies the positivity technique, via Weil's explicit formula, to control the first zeros and vanishing orders of Dirichlet $L$-functions under GRH. It delivers explicit, conditional bounds for individual characters $L(s,\chi)$ and provides sharp average results, as well as minimum/maximum zero-height and small-zero proportion results for the family modulo $q$, with extensions to general $L$-functions. The quadratic (symplectic) family is treated to obtain non-vanishing and small-zero-proportion results, including localized analyses and quantitative thresholds. Overall, the work demonstrates the versatility of the positivity framework, delivering explicit, effective bounds that can be used for computation and further theory in the distribution of low-lying zeros.

Abstract

Assuming the generalized Riemann hypothesis, we rediscover and sharpen some of the best known results regarding the distribution of low-lying zeros of Dirichlet $L$-functions. This builds upon earlier work of Omar, which relies on the classical positivity technique of explicit formulas. In addition, we generalize some of our results to a larger class of $L$-functions and provide effective conditional estimates for the lowest zeros of Dirichlet $L$-functions.

The positivity technique and low-lying zeros of Dirichlet $L$-functions

TL;DR

This paper develops and applies the positivity technique, via Weil's explicit formula, to control the first zeros and vanishing orders of Dirichlet -functions under GRH. It delivers explicit, conditional bounds for individual characters and provides sharp average results, as well as minimum/maximum zero-height and small-zero proportion results for the family modulo , with extensions to general -functions. The quadratic (symplectic) family is treated to obtain non-vanishing and small-zero-proportion results, including localized analyses and quantitative thresholds. Overall, the work demonstrates the versatility of the positivity framework, delivering explicit, effective bounds that can be used for computation and further theory in the distribution of low-lying zeros.

Abstract

Assuming the generalized Riemann hypothesis, we rediscover and sharpen some of the best known results regarding the distribution of low-lying zeros of Dirichlet -functions. This builds upon earlier work of Omar, which relies on the classical positivity technique of explicit formulas. In addition, we generalize some of our results to a larger class of -functions and provide effective conditional estimates for the lowest zeros of Dirichlet -functions.

Paper Structure

This paper contains 27 sections, 16 theorems, 136 equations, 2 figures.

Table of Contents

  1. Introduction and statement of results
  2. Individual Dirichlet -functions
  3. The family of Dirichlet -functions modulo
  4. Statement of results.
  5. Preliminaries and lemmas
  6. Weil's explicit formula
  7. Overview of the positivity technique
  8. Test functions
  9. Lemmas
  10. Estimates for individual Dirichlet -functions: proofs of Theorems \ref{['theorem gamma_chi']}--\ref{['theorem tilde gamma_chi']}
  11. Bounding : proof of Theorem \ref{['theorem gamma_chi']}
  12. Bounding : proof of Theorem \ref{['theorem n_chi']}
  13. Bounding : proof of Theorem \ref{['theorem tilde gamma_chi']}
  14. Estimates for the family of Dirichlet -functions modulo : proofs of Theorems \ref{['theorem average n_chi']}--\ref{['theorem: proportion min gamma_q']}
  15. Lemmas
  16. ...and 12 more sections

Key Result

Theorem 1

Assuming GRH, Moreover, for any constant $C>\log 4+1$, $L(s,\chi)$ has at least $(\frac{\pi^2}{8}(C-\log 4-1)+o(1))\frac{\log q}{(\log\log q)^2}$ zeros with height at most $\frac{\pi}{2}(\log\log q)^{-1}+\frac{\pi}{2}C(\log\log q)^{-2}$.

Figures (2)

  • Figure 1: A plot illustrating the lower bounds provided by \ref{['Hughes Rudnick proportion']} and Theorem \ref{['theorem: proportion min gamma_q']} for small $\beta$.
  • Figure 2: A plot of the right-hand side of \ref{['equation proportion quadratic']} for small $\beta$.

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8: Weil
  • Lemma 9
  • ...and 16 more