The Ratios Conjecture and upper bounds for negative moments of $L$-functions over function fields

TL;DR

This paper proves special cases of the Ratios Conjecture for quadratic Dirichlet L-functions over function fields, providing asymptotic formulas for averages of ratios and products, and deriving bounds for negative moments and zero density results.

Contribution

It establishes new asymptotic formulas for ratios and products of L-functions over function fields, advancing understanding of the Ratios Conjecture in this setting.

Findings

Asymptotic formulas for averages of L-function ratios over function fields.

Upper bounds for negative moments of L-functions.

Recovery of the one-level density of zeros with Fourier support in (-2,2).

Abstract

We prove special cases of the Ratios Conjecture for the family of quadratic Dirichlet $L$--functions over function fields. More specifically, we study the average of $L(1/2+\alpha,\chi_D)/L(1/2+\beta,\chi_D)$, when $D$ varies over monic, square-free polynomials of degree $2g+1$ over $\mathbb{F}_q[x]$, as $g \to \infty$, and we obtain an asymptotic formula when $\Re \beta \gg g^{-1/2+\varepsilon}$. We also study averages of products of $2$ over $2$ and $3$ over $3$ $L$--functions, and obtain asymptotic formulas when the shifts in the denominator have real part bigger than $g^{-1/4+\varepsilon}$ and $g^{-1/6+\varepsilon}$ respectively. The main ingredient in the proof is obtaining upper bounds for negative moments of $L$--functions. The upper bounds we obtain are expected to be almost sharp in the ranges described above. As an application, we recover the asymptotic formula for the one-level density of zeros in the family with the support of the Fourier transform in $(-2,2)$.