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Abstract We establish two new variants of arithmetic quantum ergodicity. The first is for self-dual$$\textrm{GL}_2$$

{GL}_{2}

Hecke–Maaß newforms over$$\mathbb {Q}$$

Q

as the level and Laplace eigenvalue vary jointly. The second is a nonsplit analogue wherein almost all restrictions of Hilbert (respectively Bianchi) Hecke–Maaß cusp forms to the modular surface dissipate as their Laplace eigenvalues grow.

A new zero-free region for Rankin–Selberg 𝐿-functions

https://doi.org/10.1515/crelle-2025-0009

Harcos, Gergely; Thorner, Jesse (March 2025, Journal für die reine und angewandte Mathematik (Crelles Journal))

Abstract Let 𝜋 and

π^{'}

\pi^{\prime}be cuspidal automorphic representations of

GL (n)

\mathrm{GL}(n)and

GL (n^{'})

\mathrm{GL}(n^{\prime})with unitary central characters.We establish a new zero-free region for all

GL (1)

\mathrm{GL}(1)-twists of the Rankin–Selberg 𝐿-function

L (s, π \times π^{'})

L(s,\pi\times\pi^{\prime}), generalizing Siegel’s celebrated work on Dirichlet 𝐿-functions.As an application, we prove the first unconditional Siegel–Walfisz theorem for the Dirichlet coefficients of

- L^{'} (s, π \times π^{'}) / L (s, π \times π^{'})

-L^{\prime}(s,\pi\times\pi^{\prime})/L(s,\pi\times\pi^{\prime}).Also, for

n \leq 8

n\leq 8, we extend the region of holomorphy and nonvanishing for the twisted symmetric power 𝐿-functions

L (s, π, {Sym}^{n} \otimes χ)

L(s,\pi,\mathrm{Sym}^{n}\otimes\chi)of any cuspidal automorphic representation of

GL (2)

\mathrm{GL}(2).

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