Negative Moments of L–Functions With Small Shifts Over Function Fields

Abstract

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 $k^{\textrm{th}}$ shifted negative moment of $L(1/2+\beta ,\chi _{D})$, in certain ranges of $\beta $ (e.g., when roughly $\beta \gg \log g/g $ and $k<1$). We also obtain non-trivial upper bounds for the $k^{\textrm{th}}$ shifted negative moment when $\log (1/\beta ) \ll \log g$. Previously, almost sharp upper bounds were obtained in [ 3] in the range $\beta \gg g^{-\frac{1}{2k}+\epsilon }$.