Alex Pascadi: Density theorems and large sieves for GL(n)

We discuss density theorems towards the Generalized Ramanujan Conjecture (GRC), as well as large sieve inequalities, for arbitrary families of GL(n) automorphic representations in the analytic-conductor aspect. The density theorems are in some cases strong enough to recover the pointwise Luo-Rudnick-Sarnak bounds towards the GRC, while the large sieve inequalities completely remove the dependency on the GRC from previous works. These results are based on new positivity properties of Rankin-Selberg L-functions, and work well in complementary ranges to spectral methods pioneered by Blomer. This is based on two papers, one joint with Jared D. Lichtman and the other joint with Jesse Thorner.