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A generalization of the Goldston-Pintz-Yildirim prime gaps result to number fields. (English) Zbl 1291.11133
Acta Math. Hung. 141, No. 1-2, 84-112 (2013); erratum ibid. 144, No. 2, 530 (2014).
Let \(p_n\) be the \(n\)th prime. A celebrated result of Goldston-Pintz-Yildirim is that \[ \liminf_{n\to\infty} \frac{p_{n+1}-p_n}{\log p_n}=0. \] In the paper under review, the author extends this result to the set-up of totally real number fields. The result is that if \({\mathbb K}\) is a totally real number field, then \[ \liminf_{_{\substack{ \omega_0,\omega_1\in {\mathcal O}_{\mathbb K}~\text{primes}\\ \omega_0\neq \omega_1}}} \frac{N_{{\mathbb K}/{\mathbb Q}}(\omega_1-\omega_0)} {N_{{\mathbb K}/{\mathbb Q}}(\omega_0)}=0. \] A key ingredient in the proof of the Goldston-Pintz-Yildirim result is the Bombieri-Vinogradiv theorem. The present work follows closely that proof with the Bombieri-Vinogradov theorem replaced by a generalization of it due to Hintz to the case of totally real number fields.

MSC:
11R80 Totally real fields
11R45 Density theorems
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