Minimal zero-free regions for results on primes between consecutive perfect $k$th powers
Acta Arithmetica
MSC: Primary 11M26; Secondary 11N05, 11L20
DOI: 10.4064/aa260304-20-7
Published online: 1 October 2026
Abstract
We compute minimal zero-free regions for the Riemann zeta-function which ensure there is always a prime between consecutive perfect $k$th powers. Our computations cover powers $k\geq 65$ 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.
