Acta Arithmetica· 2026Q2
Minimal zero-free regions for results on primes between consecutive perfect kth powers
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- 2026year
Short summary
New computations establish minimal zero-free regions for the Riemann zeta-function, proving primes exist between consecutive perfect kth powers for k≥65.
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Key points
- Minimal zero-free regions for the Riemann zeta-function are computed for k≥65.
- These regions guarantee a prime between consecutive perfect kth powers for k≥65.
- Progress towards Legendre's conjecture is quantified.
- Primes are proven to exist between consecutive 86th powers.
- An integer sequence is identified for which primes exist between consecutive 70th powers.
AI-generated from the title and abstract; the full text is not read.
Abstract
We compute minimal zero-free regions for the Riemann zeta-function which ensure there is always a prime between consecutive perfect kth powers. Our computations cover powers k≥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.
The authors' abstract, as published at the source. Acta Arithmetica, 2026 · DOI ↗
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Field: Algebra and Number Theory
Algebra and Number TheoryMathematics