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Published May 2, 2023 | Published
Journal Article Open

One-level density estimates for Dirichlet L-functions with extended support


We estimate the 1-level density of low-lying zeros of L(s,χ) with χ ranging over primitive Dirichlet characters of conductor in [Q/2,Q] and for test functions whose Fourier transform is supported in [−2−50/1093,2+50/1093]. 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 "trivial range" that follows from a simple application of the underlying trace formula (in this case orthogonality of characters). We also highlight consequences for non-vanishing of L(s,χ).

Additional Information

© 2023 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). Open Access made possible by participating institutions via Subscribe to Open. 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.

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August 22, 2023
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