Geodesic
Primes in tuples I
Daniel A. Goldston 1   ; János Pintz 2   ; Cem Y. Yíldírím 3
1 Department of Mathematics<br/>San Jose State University<br/>One Washington Square<br/>San Jose, CA 95192-0130<br/>United States
2 Alfred Rényi Institute of Mathematics<br/>Hungarian Academy of Sciences<br/> P.O. Box 127<br/>1364 Budapest<br/>Hungary
3 Feza Gürsey Enstitüsü<br/>Kuleli Mahallesi, Şekip Ayhan Özışık Caddesi 44<br/>34684 Çengelköy, İstanbul<br/>Turkey
Annals of mathematics, Tome 170 (2009) no. 2, pp. 819-862 Cet article a éte moissonné depuis la source Annals of Mathematics website

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We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that there are infinitely often primes differing by 16 or less. Even a much weaker conjecture implies that there are infinitely often primes a bounded distance apart. Unconditionally, we prove that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing, that is, \[ \liminf_{n\to \infty} \frac{p_{n+1}-p_n}{\log p_n} =0 .\] We will quantify this result further in a later paper.

DOI : 10.4007/annals.2009.170.819

Daniel A. Goldston  1   ; János Pintz  2   ; Cem Y. Yíldírím  3

1 Department of Mathematics<br/>San Jose State University<br/>One Washington Square<br/>San Jose, CA 95192-0130<br/>United States
2 Alfred Rényi Institute of Mathematics<br/>Hungarian Academy of Sciences<br/> P.O. Box 127<br/>1364 Budapest<br/>Hungary
3 Feza Gürsey Enstitüsü<br/>Kuleli Mahallesi, Şekip Ayhan Özışık Caddesi 44<br/>34684 Çengelköy, İstanbul<br/>Turkey 
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Daniel A. Goldston; János Pintz; Cem Y. Yíldírím. Primes in tuples I. Annals of mathematics, Tome 170 (2009) no. 2, pp. 819-862. doi: 10.4007/annals.2009.170.819

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