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We prove special cases of the Ratios Conjecture for the family of quadraticDirichlet $L$--functions over function fields. More specifically, we study theaverage of $L(1/2+\alpha,\chi_D)/L(1/2+\beta,\chi_D)$, when $D$ varies overmonic, square-free polynomials of degree $2g+1$ over $\mathbb{F}_q[x]$, as $g\to \infty$, and we obtain an asymptotic formula when $\Re \beta \ggg^{-1/2+\varepsilon}$. We also study averages of products of $2$ over $2$ and$3$ over $3$ $L$--functions, and obtain asymptotic formulas when the shifts inthe denominator have real part bigger than $g^{-1/4+\varepsilon}$ and$g^{-1/6+\varepsilon}$ respectively. The main ingredient in the proof isobtaining upper bounds for negative moments of $L$--functions. The upper boundswe obtain are expected to be almost sharp in the ranges described above. As anapplication, we recover the asymptotic formula for the one-level density ofzeros in the family with the support of the Fourier transform in $(-2,2)$.

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