Minimal zero-free regions for results on primes between consecutive perfect kth powers

We compute minimal zero-free regions for the Riemann zeta-function which ensure there is always a prime between consecutive perfect kth powers. Our computations cover powers k≥65 and quantify how far we are away from proving certain milestones toward an infamous open problem (Legendre’s conjecture). In addition, we prove there is always a prime between consecutive perfect 86th powers and identify an integer sequence for which there is always a prime between consecutive 70th powers.