Negative moments of $L$-functions with small shifts over function fields

Abstract

We consider negative moments of quadratic Dirichlet $L$--functions over function fields. Summing over monic square-free polynomials of degree $2g+1$ in ${\mathbb{F}}_q[x]$, we obtain an asymptotic formula for the $k^{\text{th}}$ shifted negative moment of $L(1/2+\beta ,{\chi }_D)$, in certain ranges of $\beta$ (for example, when roughly $\beta \gg \log g/g$ and $k<1$). We also obtain non-trivial upper bounds for the $k^{\text{th}}$ shifted negative moment when $\log (1/\beta )\ll \log g$. Previously, almost sharp upper bounds were obtained in \cite{ratios} in the range $\beta \gg g^{-\frac{1}{2k}+\epsilon }$.

arXiv comment27 pages; correction in the statement of Theorem 1.2