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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.

11R80 Totally real fields
11R45 Density theorems
Full Text: DOI
[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
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