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, χ _D)/L(1/2+β, χ _D), 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 ^{− 1/2+ε.} 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 ^−1/4+ε and g ^−1/6+ε 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.