Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Bounded gaps between primes in special sequences


Authors: Lynn Chua, Soohyun Park and Geoffrey D. Smith
Journal: Proc. Amer. Math. Soc. 143 (2015), 4597-4611
MSC (2010): Primary 11N05, 11N36
DOI: https://doi.org/10.1090/proc/12607
Published electronically: May 22, 2015
MathSciNet review: 3391020
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We use Maynard's methods to show that there are bounded gaps between primes in the sequence $ \{\lfloor n\alpha \rfloor \}$, where $ \alpha $ is an irrational number of finite type. In addition, given a superlinear function $ f$ satisfying some properties described by Leitmann, we show that for all $ m$ there are infinitely many bounded intervals containing $ m$ primes and at least one integer of the form $ \lfloor f(q)\rfloor $ with $ q$ a positive integer.


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

  • [1] William D. Banks and Igor E. Shparlinski, Prime numbers with Beatty sequences, Colloq. Math. 115 (2009), no. 2, 147-157. MR 2491740 (2010a:11177), https://doi.org/10.4064/cm115-2-1
  • [2] Jacques Benatar,
    The existence of small prime gaps in subsets of the integers.
    Preprint, 2014.
  • [3] Harold Davenport, Multiplicative number theory, 3rd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 2000. Revised and with a preface by Hugh L. Montgomery. MR 1790423 (2001f:11001)
  • [4] Daniel A. Goldston, János Pintz, and Cem Y. Yıldırım, Primes in tuples. I, Ann. of Math. (2) 170 (2009), no. 2, 819-862. MR 2552109 (2011c:11146), https://doi.org/10.4007/annals.2009.170.819
  • [5] Lauwerens Kuipers and Harald Niederreiter,
    Uniform distribution of sequences.
    Courier Dover Publications, 2012.
  • [6] Dieter Leitmann, The distribution of prime numbers in sequences of the form $ [f(n)]$, Proc. London Math. Soc. (3) 35 (1977), no. 3, 448-462. MR 0485743 (58 #5555)
  • [7] James Maynard, Almost-prime $ k$-tuples, Mathematika 60 (2014), no. 1, 108-138. MR 3164522, https://doi.org/10.1112/S0025579313000028
  • [8] James Maynard, Small gaps between primes, Ann. of Math. (2) 181 (2015), no. 1, 383-413. MR 3272929, https://doi.org/10.4007/annals.2015.181.1.7
  • [9] D.H.J. Polymath,
    Variants of the Selberg sieve, and bounded intervals containing many primes.
    Preprint, 2014.
  • [10] Jesse Thorner,
    Bounded gaps between primes in chebotarev sets.
    Res. Math. Sci., 1(4), 2014.
  • [11] I. M. Vinogradov,
    The method of trigonometrical sums in the theory of numbers.
    Dover Publications, Inc., Mineola, NY, 2004.
    Translated from the Russian, revised and annotated by K. F. Roth and Anne Davenport, Reprint of the 1954 translation.
  • [12] Yitang Zhang, Bounded gaps between primes, Ann. of Math. (2) 179 (2014), no. 3, 1121-1174. MR 3171761, https://doi.org/10.4007/annals.2014.179.3.7

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11N05, 11N36

Retrieve articles in all journals with MSC (2010): 11N05, 11N36


Additional Information

Lynn Chua
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachu- setts Avenue, Cambridge, Massachusetts 02139
Email: chualynn@mit.edu

Soohyun Park
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 3 Ames Street, Cambridge, Massachusetts 02139
Email: soopark@mit.edu

Geoffrey D. Smith
Affiliation: Department of Mathematics, Yale University, 10 Hillhouse Avenue, New Haven, Connecticut 06511
Email: geoffrey.smith@yale.edu

DOI: https://doi.org/10.1090/proc/12607
Received by editor(s): July 7, 2014
Received by editor(s) in revised form: July 20, 2014
Published electronically: May 22, 2015
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society