## Abstract

We estimate the 1-level density of low-lying zeros of L(s, χ) with χ ranging over primitive Dirichlet^{[ ]} characters of conductor in (Formula presented) and for test functions whose Fourier transform is supported in (Formula presented) Previously, any extension of the support past the range (−2, 2) was only known conditionally on deep conjectures about the distribution of primes in arithmetic progressions, beyond the reach of the generalized Riemann hypothesis (e.g., Montgomery’s conjecture). Our work provides the first example of a family of L-functions in which the support is unconditionally extended past the “diagonal range” that follows from a straightforward application of the underlying trace formula (in this case orthogonality of characters). We also highlight consequences for nonvanishing of L(s, χ).

Original language | English (US) |
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Pages (from-to) | 805-830 |

Number of pages | 26 |

Journal | Algebra and Number Theory |

Volume | 17 |

Issue number | 4 |

DOIs | |

State | Published - 2023 |

### Funding

Part of this work was conducted while Pratt was supported by the National Science Foundation Graduate Research Program under grant number DGE-1144245. Radziwiłł acknowledges the support of a Sloan fellowship and NSF grant DMS-1902063. The authors thank the referee for helpful remarks, and Jared Lichtman for helpful discussions on Proposition 6.

## Keywords

- Dirichlet L-functions
- arithmetic progressions
- dispersion method
- nonvanishing
- one-level density
- primes

## ASJC Scopus subject areas

- Algebra and Number Theory