An unconditional $\mathrm{GL}(n)$ large sieve

Abstract

Let ${\mathfrak{F}}_n$ be the set of all cuspidal automorphic representations $\pi$ of ${\mathrm{GL}}_n$ over a number field with unitary central character. We prove two unconditional large sieve inequalities for the Hecke eigenvalues of $\pi \in {\mathfrak{F}}_n$, one on the integers and one on the primes. The second leads to the first unconditional zero density estimate for the family of $L$-functions $L(s,\pi )$ associated to $\pi \in {\mathfrak{F}}_n$, which we make log-free. As an application of the zero density estimate, we prove a hybrid subconvexity bound for $L(\frac{1}{2},\pi )$ for a density one subset of $\pi \in {\mathfrak{F}}_n$.

arXiv comment17 pages. Incorporates referee comments

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Cited by in corpus (5)

• A zero density estimate for Dedekind zeta functions
• An approximate form of Artin's holomorphy conjecture and non-vanishing of Artin $L$-functions
• Zeros of Rankin-Selberg $L$-functions in families
• On the spectral large sieve inequality for symmetric-squares
• Effective quantum unique ergodicity for Hecke-Maass newforms and Landau-Siegel zeros