Prime-free discs in imaginary quadratic fields
Tanmay Khale
TL;DR
The paper proves that for imaginary quadratic fields K, the maximal prime-free ball radius G_K(X) satisfies G_K(X) ≫_K ( olinebreak n lacksquare) ( olinebreak log X) rac{ olinebreak log_2(X) olinebreak log_4(X)}{ olinebreak log_3(X)}. The authors develop a number-field analogue of dense-cluster sieve methods, combining ray-class Hecke L-functions, Page–Bombieri–Vinogradov machinery, and finely tuned sieve weights with a two-stage random selection and a hypergraph covering argument. Key contributions include a generalization of Q-field hole bounds to imaginary quadratic fields, a robust sieve framework in number fields, and a probabilistic covering approach that achieves near-optimal hole sizes. Overall, the work advances prime-distribution results in imaginary quadratic fields and provides versatile tools for prime gaps via a synthesis of analytic, algebraic, and combinatorial techniques.
Abstract
Suppose $K$ is an imaginary quadratic field, and let $N_K$ denote the field norm in the ring of integers $O_K$. Let $B(x_0,r) = \{x \in O_K: |N_K(x-x_0)| < r\}$. Let $G_K(X) = \max \{r > 0: \text{there exists } x_0 \in O_K \text{ such that } |N_K(x_0)| \leq X \text{ and } B(x_0,r) \text{ contains no primes} \}$. We show that $ G_{K}(X) \gg_K (\log X) \frac{\log_2(X) \log_4(X)}{\log_3(X)}$.