Expected values of cubic Dirichlet $L$-functions away from the central point
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- by Chantal David and Patrick Meisner;
- Trans. Amer. Math. Soc. 378 (2025), 5125-5157
- DOI: https://doi.org/10.1090/tran/9428
- Published electronically: March 25, 2025
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Abstract:
We compute the expected value of Dirichlet $L$-functions over $\mathbb {F}_q[T]$ attached to cubic characters evaluated at an arbitrary $s \in (0,1)$. We find a transition term at the point $s=\frac {1}{3}$, reminiscent of the transition at the point $s=\frac {1}{2}$ of the bound for the size of an $L$-function implied by the Lindelöf hypothesis. We show that at $s=\frac {1}{3}$, the expected value matches corresponding statistics of the group of unitary matrices multiplied by a weight function. This is the first result in the literature computing the first moment at $s=\tfrac 13$ for any family of cubic Dirichlet characters, over function fields or number fields, and it involves the deep connections between Dirichlet series of cubic Gauss sums and metaplectic Eisenstein series first introduced by Kubota, which is necessary to obtain the cancellation between the principal sum and the dual sum occurring at $s=\tfrac 13$.References
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Bibliographic Information
- Chantal David
- Affiliation: Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve West, Montréal, Québec, H3G 1M8, Canada
- MR Author ID: 363267
- Email: chantal.david@concordia.ca
- Patrick Meisner
- Affiliation: Department of Mathematical Sciences, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
- MR Author ID: 1204935
- Email: meisner@chalmers.se
- Received by editor(s): October 30, 2023
- Received by editor(s) in revised form: January 21, 2025
- Published electronically: March 25, 2025
- Additional Notes: The first author was supported by the Natural Sciences and Engineering Research Council of Canada [DG-155635-2019] and by the Fonds de recherche du Québec Nature et technologies [Projet de recherche en équipe 300951]. The second author was supported by the grant KAW 2019.0517 from the Knut and Alice Wallenberg Foundation
- © Copyright 2025 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 378 (2025), 5125-5157
- MSC (2020): Primary 11M06, 11M38, 11R16, 11R58
- DOI: https://doi.org/10.1090/tran/9428