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+alpha, XD)/L(1/2+Q, XD), when D varies over monic, square-free polynomials of degree 2g +1 over Fq[x], as g -> infinity, and we obtain an asymptotic formula when Q >> g-1/2+epsilon. 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+epsilon and g-1/6+epsilon 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.