The Ratios Conjecture and upper bounds for negative moments of $L$-functions over function fields

We prove special cases of the Ratios Conjecture for the family of quadratic Dirichlet L–functions over function fields. More specifically, we study the average of L(1/2 + α, χD)/L(1/2 + β, χD), when D varies over monic, square-free polynomials of degree 2g + 1 over Fq[x], as g → ∞, and we obtain an asymptotic formula when Rβ ≫ g. We also study averages of products of 2 over 2 and 3 over 3 L– functions, and obtain asymptotic formulas when the shifts in the denominator have real part bigger than g and g respectively. The main ingredient in the proof is obtaining upper bounds for negative moments of L–functions. The upper bounds we obtain are expected to be almost sharp in the ranges described above. As an application, we recover the asymptotic formula for the one-level density of zeros in the family with the support of the Fourier transform in (−2, 2).

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