Skip to main content

Distribution of Large Gaps Between Primes

  • Chapter
  • First Online:
Irregularities in the Distribution of Prime Numbers


We survey some past conditional results on the distribution of large gaps between consecutive primes and examine how the Hardy–Littlewood prime k-tuples conjecture can be applied to this question.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD 54.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 99.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others


  1. 1.

    Selberg also proved an unconditional estimate that we are not concerned with in this paper.

  2. 2.

    Recent work [1] has determined that C = 0.84 is acceptable.


  1. E. Carneiro, M.B. Milinovich, K. Soundararajan, Fourier optimization and prime gaps (submitted for publication). Available at

  2. H. Cramér, Some theorems concerning prime numbers. Arkiv för Mat. Astr. och Fys. 15(5), 1–32 (1920)

    Google Scholar 

  3. H. Cramér, Prime numbers and probability. Skand. Mat. Kongr. 8, 107–115 (1935)

    MATH  Google Scholar 

  4. H. Cramér, On the order of magnitude of the difference between consecutive prime numbers. Acta Arith. 2, 23–46 (1936)

    Article  Google Scholar 

  5. P.X. Gallagher, Some consequences of the Riemann hypothesis. Acta Arith. 37, 339–343 (1980)

    Article  MathSciNet  Google Scholar 

  6. P.X. Gallagher, On the distribution of primes in short intervals. Mathematika 23, 4–9 (1976); Corrigendum: Mathematika 28(1), 86 (1981)

    Article  MathSciNet  Google Scholar 

  7. P.X. Gallagher, J.H. Mueller, Primes and zeros in short intervals. J. Reine Angew. Math. 303/304, 205–220 (1978)

    Google Scholar 

  8. D.A. Goldston, D.R. Heath-Brown, A note on the difference between consecutive primes. Math. Annalen 266, 317–320 (1984)

    Article  MathSciNet  Google Scholar 

  9. D.A. Goldston, A.H. Ledoan, On the Differences Between Consecutive Prime Numbers, I, Integers 12B (2012/13), 8 pp. [Paper No. A3]. Also appears in: Combinatorial Number Theory (Proceedings of the “Integers Conference 2011,” Carrollton, Georgia, October 26–29, 2011), De Gruyter Proceedings in Mathematics (2013), pp. 37–44

    Google Scholar 

  10. D.A. Goldston, H.L. Montgomery, Pair correlation of zeros and primes in short intervals, in Analytic Number Theory and Diophantine Problems (Birkhauser, Boston, MA, 1987), pp. 183–203

    Google Scholar 

  11. A. Granville, Harald Cramér and the distribution of prime numbers, Harald Cramér Symposium (Stockholm, 1993). Scand. Actuar. J. 1, 12–28 (1995)

    Article  Google Scholar 

  12. H. Halberstam, H.-E. Richert, Sieve Methods. London Mathematical Society Monographs, vol. 4 (Academic Press, London/New York/San Francisco, 1974)

    Google Scholar 

  13. G.H. Hardy, J.E. Littlewood, Some problems of ‘Partitio numerorum’; III: on the expression of a number as a sum of primes. Acta Math. 44(1), 1–70 (1922). Reprinted in Collected Papers of G.H. Hardy, vol. I (including joint papers with J.E. Littlewood and others; edited by a committee appointed by the London Mathematical Society) (Clarendon Press/Oxford University Press, Oxford, 1966), pp. 561–630

    Google Scholar 

  14. D.R. Heath-Brown, Gaps between primes, and the pair correlation of zeros of the zeta-function. Acta Arith. 41, 85–99 (1982)

    Article  MathSciNet  Google Scholar 

  15. H. Maier, Primes in short intervals. Mich. Math. J. 32(2), 221–225 (1985)

    Article  MathSciNet  Google Scholar 

  16. H.L. Montgomery, The pair correlation of zeros of the zeta function, in Analytic Number Theory, Proceedings of Symposia in Pure Mathematics, vol. XXIV (St. Louis University, St. Louis, MO, 1972), pp. 181–193; (American Mathematical Society, Providence, RI, 1973)

    Google Scholar 

  17. H.L. Montgomery, Early Fourier Analysis. Pure and Applied Undergraduate Texts, vol. 22 (American Mathematical Society, Providence, RI, 2014)

    Google Scholar 

  18. H.L. Montgomery, K. Soundararajan, Primes in short intervals. Commun. Math. Phys. 252(1–3), 589–617 (2004)

    Article  MathSciNet  Google Scholar 

  19. J.H. Mueller, On the difference between consecutive primes, in Recent Progress in Analytic Number Theory, Durham, vol. 1 (1979), pp. 269–273 (Academic, London/New York, 1981)

    Google Scholar 

  20. B. Saffari, R.C. Vaughan, On the fractional parts of xn and related sequences, II. Ann. Inst. Fourier (Grenoble) 27(2), 1–30 (1977)

    Google Scholar 

  21. A. Selberg, On the normal density of primes in small intervals, and the difference between consecutive primes. Arch. Math. Naturvid. 47(6), 87–105 (1943)

    MathSciNet  MATH  Google Scholar 

Download references


The authors wish to express their sincere gratitude and appreciation to the anonymous referee for carefully reading the original version of this paper and for making a number of very helpful comments and suggestions.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Daniel A. Goldston .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Funkhouser, S., Goldston, D.A., Ledoan, A.H. (2018). Distribution of Large Gaps Between Primes. In: Pintz, J., Rassias, M. (eds) Irregularities in the Distribution of Prime Numbers. Springer, Cham.

Download citation

Publish with us

Policies and ethics