Minimal zero-free regions for results on primes between consecutive perfect $k$th powers

We compute minimal zero-free regions for the Riemann zeta-function of the Littlewood form which ensure there is always a prime between consecutive perfect $k$th powers. Our computations cover powers $k\ge 65$ 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 $86$th powers, and identify an integer sequence (that is a subset of the positive integers) for which there is always a prime between consecutive $70$th powers.