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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 GL  d   L -functions.

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