Skip to main content
SpringerLink
Log in

Unexpected Irregularities in the Distribution of Prime Numbers

  • Conference paper
Book cover Proceedings of the International Congress of Mathematicians

Abstract

In 1849 the Swiss mathematican ENCKE wrote to GAUSS, asking whether he had ever considered trying to estimate Π(x), the number of primes up to x, by some sort of “smooth” function. On Christmas Eve 1849, GAUSS replied that “he had pondered this problem as a boy” and had come to the conclusion that “at around x, the primes occur with density 1/log x.” Thus, he concluded, π(ϰ) could be approximated by

$$Li(x): = \int_2^x {\frac{{dt}}{{\log t}} = \frac{x}{{\log x}} + \frac{x}{{{{\log }^2}x}} + O(\frac{x}{{{{\log }^3}x}})}$$

.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Balog, The prime k-tuplets conjecture on average, Analytic Number Theory (B.C. Berndt, H.G. Diamond, H. Halberstam, A. Hildebrand, eds.), Basel and Birkhäuser, Boston, 1990, pp. 165–204.

    Google Scholar 

  2. E. Bombieri, J. B. Friedlander, and H. Iwaniec, Primes in arithmetic progressions to large moduli III, J. Amer. Math. Soc. 2 (1989), 215–224.

    Article  MathSciNet  Google Scholar 

  3. A. A. Buchstab, On an asymptotic estimate of the number of numbers of an arithmetic progression which are not divisible by relatively small prime numbers, Mat. Sb. 28/70 (1951), 165–184 (Russian).

    MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  5. H. Davenport, Multiplicative Number Theory (2nd ed.), Springer-Verlag, Berlin and New York, 1980.

    Book  Google Scholar 

  6. J. B. Friedlander and A. Granville, Limitations to the equi-distribution of primes I, Ann. of Math. (2) 129 (1989), 363–382.

    Article  MathSciNet  Google Scholar 

  7. J. B. Friedlander and A. Granville, Limitations to the equi-distribution of primes IV, Proc. Royal Soc. London Ser. A 435 (1991), 197–204.

    Article  MathSciNet  Google Scholar 

  8. J. B. Friedlander and A. Granville, Limitations to the equi-distribution of primes III, Compositio Math. 81 (1992), 19–32.

    MathSciNet  MATH  Google Scholar 

  9. J. B. Friedlander and A. Granville, Relevance of the residue class to the abundance of primes, Proc. Amalfi Conf. on Analytic Number Theory (E. Bombieri, A. Perelli, S. Salerno, U. Zannier, eds.), Salerno, Italy, 1993, pp. 95–104.

    Google Scholar 

  10. J. B. Friedlander, A. Granville, A. Hildebrand, and H. Maier, Oscillation theorems for primes in arithmetic progressions and for sifting functions, J. Amer. Math. Soc. 4 (1991), 25–86.

    Article  MathSciNet  Google Scholar 

  11. P. X. Gallagher, On the distribution of primes in short intervals, Mathematika 23 (1976), 4–9.

    Article  MathSciNet  Google Scholar 

  12. D. A. Goldston, On the pair correlation conjecture for zeros of the Riemann zeta-function, J. Reine Angew. Math. 385 (1988), 24–40.

    MathSciNet  MATH  Google Scholar 

  13. A. Granville, Harald Cramer and the distribution of prime numbers, Scand. Actuarial J. 2 (1995), to appear.

    Google Scholar 

  14. G. H. Hardy and J. E. Littlewood, Some problems on partitio numerorum III On the expression of a number as a sum of primes, Acta Math. 44 (1923), 1–70.

    Article  MathSciNet  Google Scholar 

  15. A. Hildebrand and H. Maier, Irregularities in the distribution of primes in short intervals, J. Reine Angew. Math. 397 (1989), 162–193.

    MathSciNet  MATH  Google Scholar 

  16. H. Maier, Primes in short intervals, Michigan Math. J. 32 (1985), 221–225.

    Article  MathSciNet  Google Scholar 

  17. H. L. Montgomery, The pair correlation of zeros of the zeta function, Proc. Sympos. Pure Math., vol. 24, Amer. Math. Soc., Providence, RI, 1973, pp. 181–193.

    Article  MathSciNet  Google Scholar 

  18. M. Nair and A. Perelli, On the prime ideal theorem and irregularities in the distribution of primes, Duke Math. J. 77 (1995), 1–20.

    Article  MathSciNet  Google Scholar 

  19. A. M. Odlyzko, The 10 20 th zero of the Riemann zeta function and 70 million of its neighbors, to appear.

    Google Scholar 

  20. M. Rubinstein and P. Sarnak, Chebyshev’s bias, to appear.

    Google Scholar 

  21. Z. Rudnick and P. Sarnak, The n-level correlations of the zeros of the zetafunction, C.R. Acad. Sci. Paris, to appear.

    Google Scholar 

  22. A. Schinzel and W. Sierpinski, Sur certaines hypotheses concernant les nombres premiers, Acta Arith. 4 (1958), 185–208.

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Georgia, Athens, Georgia, 30602, USA

    Andrew Granville

Authors
  1. Andrew Granville
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Département de Mathématiques, EPFL, 1015, Laussane, Switzerland

    S. D. Chatterji

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Birkhäser Verlag, Basel, Switzerland

About this paper

Cite this paper

Granville, A. (1995). Unexpected Irregularities in the Distribution of Prime Numbers. In: Chatterji, S.D. (eds) Proceedings of the International Congress of Mathematicians. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-9078-6_32

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-9078-6_32

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9897-3

  • Online ISBN: 978-3-0348-9078-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation