Low-lying zeros of quadratic Dirichlet $L$-functions: A transition in the Ratios Conjecture
Abstract
We study the $1$-level density of low-lying zeros of quadratic Dirichlet $L$-functions by applying the $L$-functions Ratios Conjecture. We observe a transition in the main term as was predicted by the Katz-Sarnak heuristic as well as in the lower order terms when the support of the Fourier transform of the corresponding test function reaches the point $1$. Our results are consistent with those obtained in previous work under GRH and are furthermore analogous to results of Rudnick in the function field case.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2017
- DOI:
- 10.48550/arXiv.1710.06834
- arXiv:
- arXiv:1710.06834
- Bibcode:
- 2017arXiv171006834F
- Keywords:
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- Mathematics - Number Theory;
- 11M26;
- 11M50 (primary)
- E-Print:
- 15 pages