Expected values of cubic Dirichlet 𝐿-functions away from the central point

David, Chantal; Meisner, Patrick

doi:10.1090/tran/9428

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 $s=\tfrac 13$.

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Expected values of cubic Dirichlet $L$-functions away from the central point
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Abstract: