Log-free zero density estimates for automorphic $L$-functions
We prove a log-free zero density estimate for automorphic $L$-functions
defined over a number field $k$. This work generalizes and sharpens the method
of pseudo-characters and the large sieve used earlier by Kowalski and Michel.
As applications, we demonstrate for a particular family of number fields of
degree $n$ over $k$ (for any $n$) that an effective Chebotarev density theorem
and a bound on $\ell$-torsion in class groups hold for almost all fields in the