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arXiv · 2602.14340

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

Ethan Simpson Lee
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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.

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Ethan Simpson Lee. 2026-02-15. Minimal zero-free regions for results on primes between consecutive perfect $k$th powers. https://arxiv.org/abs/2602.14340

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