## Abstract

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+α, χ
_{D})/L(1/2+β, χ
_{D}), when D varies over monic, square-free polynomials of degree 2g+1 over Fq[x], as g → ∞, and we obtain an asymptotic formula when ℜβ ≫ g
^{−}
^{1/2+ε.} 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+ε} and g
^{−1/6+ε} 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.

Original language | English |
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Pages (from-to) | 4453-4510 |

Journal | Transactions of the American Mathematical Society |

Volume | 376 |

Issue number | 6 |

DOIs | |

Publication status | Published - 1 Jun 2023 |