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

2021/11/19 by Alexandra Florea, Florea, Alexandra
Mathematics · Social Sciences · #Analytic Number Theory Research #Historical Geopolitical and Social Dynamics #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2111.10477

Abstract

We consider negative moments of quadratic Dirichlet L--functions over function fields. Summing over monic square-free polynomials of degree 2g+1 in \mathbbF_q[x], we obtain an asymptotic formula for the k^th shifted negative moment of L(1/2+β,χ_D), in certain ranges of β (for example, when roughly β≫ log g/g and k<1). We also obtain non-trivial upper bounds for the k^th shifted negative moment when log(1/β) ≪ log g. Previously, almost sharp upper bounds were obtained in \citeratios in the range β≫ g^-(1)/(2k)+ε.

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