## Abstract

We study the 1-level density and the pair correlation of zeros of quadratic Dirichlet L-functions in function fields, as we average over the ensemble H
_{2g+1} of monic, square-free polynomials with coefficients in F
_{q}[x]. In the case of the 1-level density, when the Fourier transform of the test function is supported in the restricted interval (1/3, 1), we compute a secondary term of 3 size q
^{−4g/3}/g, which is not predicted by the Ratios Conjecture. Moreover, when the support is even more restricted, we obtain several lower order terms. For example, if the Fourier transform is supported in (1/3, 1/2), we identify another 2 lower order term of size q
^{−8g/5}/g. We also compute the pair correlation, and as for the 1-level density, we detect lower order terms under certain restrictions; for example, we see a term of size q
^{−g}/g
^{2} when the Fourier transform is supported in (14, 1). The 1-level density and the pair correlation allow us 2 to obtain non-vanishing results for L(1/2, χD), as well as lower bounds for the proportion of simple zeros of this family of L-functions.

Original language | English |
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Pages (from-to) | 8013-8045 |

Number of pages | 33 |

Journal | Transactions of the American Mathematical Society |

Volume | 370 |

Issue number | 11 |

Early online date | 7 Jun 2018 |

DOIs | |

Publication status | Published - Nov 2018 |