Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Gaps between prime numbers


Authors: Adolf Hildebrand and Helmut Maier
Journal: Proc. Amer. Math. Soc. 104 (1988), 1-9
MSC: Primary 11N05
MathSciNet review: 958032
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {d_n} = {p_{n + 1}} - {p_n}$ denote the $ n$th gap in the sequence of primes. We show that for every fixed integer $ k$ and sufficiently large $ T$ the set of limit points of the sequence $ \{ ({d_n}/\log n, \ldots ,{d_{n + k - 1}}/\log n)\} $ in the cube $ {[0,T]^k}$ has Lebesgue measure $ \geq c(k){T^k}$, where $ c(k)$ is a positive constant depending only on $ k$. This generalizes a result of Ricci and answers a question of Erdös, who had asked to prove that the sequence $ \{ {d_n}/\log n\} $ has a finite limit point greater than 1.


References [Enhancements On Off] (What's this?)

  • [1] P. Erdös, On the difference between consecutive primes, Quart. J. Oxford 6 (1935), 124-128.
  • [2] P. Erdös, Problems and results on the differences of consecutive primes, Publ. Math. Debrecen 1 (1949), 33–37. MR 0030994 (11,84a)
  • [3] P. X. Gallagher, A large sieve density estimate near 𝜎=1, Invent. Math. 11 (1970), 329–339. MR 0279049 (43 #4775)
  • [4] H. Halberstam and H.-E. Richert, Sieve methods, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York, 1974. London Mathematical Society Monographs, No. 4. MR 0424730 (54 #12689)
  • [5] Helmut Maier, Chains of large gaps between consecutive primes, Adv. in Math. 39 (1981), no. 3, 257–269. MR 614163 (82g:10061), http://dx.doi.org/10.1016/0001-8708(81)90003-7
  • [6] -, Small differences between prime numbers, Preprint.
  • [7] R. A. Rankin, The difference between consecutive prime numbers, J. London Math. Soc. 13 (1938), 242-247.
  • [8] Giovanni Ricci, Recherches sur l’allure de la suite {𝑝_{𝑛+1}-𝑝_{𝑛}/log𝑝_{𝑛}}, Colloque sur la Théorie des Nombres, Bruxelles, 1955, Georges Thone, Liège; Masson and Cie, Paris, 1956, pp. 93–106 (French). MR 0079605 (18,112e)
  • [9] E. Westzynthius, Über die Verteilung der Zahlen, die zu den n ersten Primzahlen teilerfremd sind, Comm. Phys. Math. Helsingfors 25 (1931), 1-37.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11N05

Retrieve articles in all journals with MSC: 11N05


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0958032-5
PII: S 0002-9939(1988)0958032-5
Article copyright: © Copyright 1988 American Mathematical Society