$\mathrm{GL}_n$ large sieves and density estimates via positive semi-definiteness

Let $\mathfrak{F}_n$ be the set of unitary cuspidal automorphic representations of $\mathrm{GL}_n$ over a number field $F$, and let $\mathcal{S}\subseteq\mathfrak{F}_n$ be a finite subset. Given $π_0\in\mathfrak{F}_{n_0}$, we establish large sieve inequalities for the families $\{L(s,π)\colon π\in\mathcal{S}\}$ and $\{L(s,π\timesπ_0)\colon π\in\mathcal{S}\}$ that, unlike previous results, are independent of progress towards the generalized Ramanujan conjecture and simultaneously handle the Dirichlet coefficients of $L$, $L^{-1}$, and $\log L$. Our approach is based on the duality principle and is sharp in ranges that are complementary to large sieve inequalities based on trace formulae. We apply our large sieve inequalities to establish several density estimates for families of $L$-functions, counting potential violations of the generalized Riemann hypothesis and the generalized Ramanujan conjecture. In particular, we remove all restrictions in the log-free zero density estimate of Brumley, Thorner, and Zaman for families of Rankin-Selberg $L$-functions.