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

We consider negative moments of quadratic Dirichlet L-functions over function fields. Summing over monic square-free polynomials of degree 2g + 1 in F-q[x], we obtain an asymptotic formula for the k(th) shifted negative moment of L (1/2+ beta, chi(D)), in certain ranges of beta(e.g., when roughly beta >> log g/g and k < 1). We also obtain non-trivial upper bounds for the k(th) shifted negative moment when log(1/beta) << log g. Previously, almost sharp upper bounds were obtained in [3] in the range beta >> g(- 1/2k+is an element of).