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

TL;DR

This paper establishes minimal zero-free regions for the Riemann zeta-function to guarantee primes between consecutive perfect powers, advancing understanding of prime distribution related to Legendre's conjecture.

Contribution

It computes new zero-free regions for the zeta-function for powers $k \,\geq\, 65$, and proves primes exist between certain consecutive perfect powers, including the 86th and a subset related to the 70th powers.

Findings

Confirmed primes between consecutive perfect 86th powers.

Identified a subset of integers with primes between consecutive 70th powers.

Quantified the gap to Milestone progress toward Legendre's conjecture.

Abstract

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 $k\geq 65$ 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 $86$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.