Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Acta Arithmetica / All issues

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.