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 Fq[x], we obtain an asymptotic formula for the k shifted negative moment of L(1/2+β, χD), when the shift β in the denominator satisfies Rβ ≥ c0 log g/g for an explicit constant c0 which depends on k, when g → ∞. We also obtain non-trivial upper bounds for the k shifted negative moment when log(1/β) ≪ log g. Previously, almost sharp upper bounds were obtained in [BFK] in the range β ≫ g− 1 2k .

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