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

Abstract

We prove special cases of the Ratios Conjecture for the family of quadratic Dirichlet L L -functions over function fields. More specifically, we study the average of L ( 1 / 2 + α , χ D ) / L ( 1 / 2 + β , χ D ) L(1/2+\alpha ,\chi _D)/L(1/2+\beta ,\chi _D) , when D D varies over monic, square-free polynomials of degree 2 g + 1 2g+1 over F q [ x ] \mathbb {F}_q[x] , as g → ∞ g \to \infty , and we obtain an asymptotic formula when ℜ β ≫ g − 1 / 2 + ε \Re \beta \gg g^{-1/2+\varepsilon } . We also study averages of products of 2 2 over 2 2 and 3 3 over 3 3 L L -functions, and obtain asymptotic formulas when the shifts in the denominator have real part bigger than g − 1 / 4 + ε g^{-1/4+\varepsilon } and g − 1 / 6 + ε g^{-1/6+\varepsilon } respectively. The main ingredient in the proof is obtaining upper bounds for negative moments of L L -functions. The upper bounds we obtain are expected to be almost sharp in the ranges described above.