Effective Log-Free Zero Density Estimates for Automorphic L-Functions and the Sato–Tate Conjecture

Let $K/\mathbb{Q}$ be a number field. Let π and π′ be cuspidal automorphic representations of $\textrm{GL}_{d}(\mathbb{A}_{K})$ and $\textrm{GL}_{d^{\prime }}(\mathbb{A}_{K})$. We prove an unconditional and effective log-free zero density estimate for all automorphic L-functions L(s, π) and prove a similar estimate for Rankin–Selberg L-functions L(s, π × π′) when π or π′ satisfies the Ramanujan conjecture. As applications, we make effective Moreno’s analog of Hoheisel’s short interval prime number theorem and extend it to the context of the Sato–Tate conjecture; additionally, we bound the least prime in the Sato–Tate conjecture in analogy with Linnik’s theorem on the least prime in an arithmetic progression. We also prove effective log-free density estimates for automorphic L-functions averaged over twists by Dirichlet characters, which allows us to prove an “average Hoheisel” result for GLdL-functions.