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,\chi_D)/L(1/2+\beta,\chi_D), when D varies over monic, square-free polynomials of degree 2g+1 over F_q[x], as g\to\infty, and we obtain an asymptotic formula when \Re(\beta) \gg g^{-1/2+\varepsilon}. 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+\varepsilon} and g^{-1/6+\varepsilon} 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.