This paper considers the Ratios Conjecture of Conrey, Farmer and Zirnbauer [B. Conrey et al., Commun. Number Theory Phys. 2, No. 3, 593–636 (2008; Zbl 1178.11056)] in the context of \(L\)-series defined over a fixed polynomial ring \(\mathbb{F}_q[x]\). We let \(\mathcal{H}_{2g+1}\) be the ensemble of monic, square-free polynomials of degree \(2g+1\), and if \(D\in\mathcal{H}_{2g+1}\) we write \(\chi_D\) for the quadratic character modulo \(D\). In this context the Ratios Conjecture predicts the asymptotic behaviour, as \(g\to\infty\), of \[ \frac{1}{|\mathcal{H}_{2g+1}|}\sum_{D\in\mathcal{H}_{2g+1}} \frac{\prod_{\alpha\in\mathcal{A}}L(\tfrac12+\alpha,\chi_D)} {\prod_{\beta\in\mathcal{B}}L(\tfrac12+\beta,\chi_D)}, \] for given sets \(\mathcal{A}=\{\alpha_1,\dots,\alpha_k\}\) and \(\mathcal{B}=\{\beta_1,\dots,\beta_k\}\) satisfying suitable constraints. In particular it is natural to assume that \(\mathrm{Re}(\beta_j)\gg g^{-1}\), so that \(\tfrac12+\beta_j\) is not too close to a zero of \(L(s,\chi_D)\).
The paper succeeds in proving the conjectured asymptotic formula, saving a power of \(q^g\), when \(k=1,2\) or 3. However it is necessary to impose further restrictions on the sets \(\mathcal{A}\) and \(\mathcal{B}\). In particular one requires a lower bound \(\mathrm{Re}(\beta_j)\gg g^{-1/2k+\varepsilon}\) for each \(j\). The hardest part of the argument lies in controlling negative moments of \(L(s,\chi_D)\). This is achieved by adapting the ideas of K. Soundararajan [Ann. Math. (2) 170, No. 2, 981–993 (2009; Zbl 1251.11058)] and A. J. Harper [Sharp conditional bounds for moments of the Riemann zeta function”, Preprint, arXiv:1305.4618], who handled the Riemann Zeta-function subject to the Riemann Hypothesis. Thus for example it is shown for any fixed \(m>\tfrac12\) that \[ \frac{1}{|\mathcal{H}_{2g+1}|}\sum_{D\in\mathcal{H}_{2g+1}} \frac{1}{|L(\tfrac12+\beta,\chi_D)|^m}\ll\beta^{-m(m-1)/2}(\log g)^{m(m+1)/2}, \] for \(\beta\in(0,\tfrac12)\) with \(\beta\gg g^{-1/2m+\varepsilon}\).