Prime-free discs in imaginary quadratic fields

2025/10/15 by Khale, Tanmay
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2510.14006

Abstract

Suppose K is an imaginary quadratic field, and let N_K denote the field norm in the ring of integers O_K. Let B(x₀,r) = \x ∈ O_K: |N_K(x-x₀)| < r\. Let G_K(X) = max \r > 0: there exists x₀ ∈ O_K such that |N_K(x₀)| ≤ X and B(x₀,r) contains no primes \. We show that G_K(X) ≫_K (log X) (log₂(X) log₄(X))/(log₃(X)).

Citations

• Large gaps between primes
• Dense clusters of primes in subsets
• Bounded gaps between primes in number fields and function fields

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