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

## TL;DR

This paper establishes new unconditional large sieve inequalities for automorphic representations of GL(n), leading to zero density estimates and subconvexity bounds for associated L-functions, advancing understanding in automorphic forms and number theory.

## Contribution

It introduces the first unconditional large sieve inequalities for GL(n) automorphic representations on integers and primes, and derives zero density and subconvexity results.

## Key findings

- Unconditional large sieve inequalities for Hecke eigenvalues on integers and primes.
- First unconditional zero density estimate for GL(n) automorphic L-functions.
- Hybrid subconvexity bounds for L(1/2, π) for a density one subset of representations.

## 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 $\pi\in\mathfrak{F}_n$.