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Patrick Meisner: Expected Values of $L$-Functions over $\mathbb{F}_q(T)$ away from the Central Point

Time: Wed 2023-05-17 11.15

Location: KTH, 3418

Participating: Patrick Meisner (Chalmers)

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I will discuss bounds on the expected value of \(L(s,\chi)\) as \(\chi\) runs over cubic Dirichlet characters defined over \(\mathbb{F}_q(T)\) for an arbitrary \(s \in (0,1)\). Specifically, one finds a transition term at the point \(s=\frac{1}{3}\), reminiscent of the transition at the point s=1/2 of the bound for the size of an L-function implied by the Lindelöf hypothesis. I will then show that this transition at s=1/3 corresponding statistics of the group of unitary matrices multiplied by a weight function. This is motivated by a result connecting weighted n-level density rules to corresponding statistics of weighted unitary matrices.