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arXiv · 1804.06402

Zeros of Rankin-Selberg $L$-functions at the edge of the critical strip

Farrell Brumley·Jesse Thorner·Asif Zaman
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Abstract

Let $\pi$ and $\pi_0$ be unitary cuspidal automorphic representations. We prove log-free zero density estimates for Rankin-Selberg $L$-functions of the form $L(s,\pi\times\pi_0)$, where $\pi$ varies in a given family and $\pi_0$ is fixed. These estimates are unconditional in many cases of interest; they hold in full generality assuming an average form of the generalized Ramanujan conjecture. We consider applications of these estimates related to mass equidistribution for Hecke-Maass forms, the rarity of Landau-Siegel zeros of Rankin-Selberg $L$-functions, the Chebotarev density theorem, and $\ell$-torsion in class groups of number fields.

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Farrell Brumley, Jesse Thorner, Asif Zaman. 2018-04-17. Zeros of Rankin-Selberg $L$-functions at the edge of the critical strip. https://doi.org/10.4171/jems/1134

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