# zbMATH — the first resource for mathematics

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
##### Keywords:
distribution of primes; algebraic number field
Full Text:
##### References:
 [1] Gallagher, P. X., On the distribution of primes in short intervals, Mathematika, 23, 4-9, (1976) · Zbl 0346.10024 [2] Goldston, D. A.; Motohashi, Y.; Pintz, J.; Yıldırım, C. Y., Small gaps between primes exist, Proc. Japan Acad., 82A, 61-65, (2006) · Zbl 1168.11041 [3] Goldston, D. A.; Pintz, J.; Yıldırım, C. Y., Primes in tuples I, Ann. of Math., 170, 819-862, (2009) · Zbl 1207.11096 [4] Hinz, J., A generalization of bombieri’s prime number theorem to algebraic number fields, Acta Arithmetica, 51, 173-193, (1988) · Zbl 0605.10023 [5] Hinz, J., Character sums in algebraic number fields, J. Number Theory, 17, 52-70, (1983) · Zbl 0511.10028 [6] Mitsui, T., Generalized prime number theorem, Japanese J. Math., 26, 1-42, (1956) · Zbl 0126.27503 [7] H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory I. Classical Theory (Cambridge, 2007). · Zbl 1142.11001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.