Skip to main content

Part of the book series: NATO Science Series ((NAII,volume 237))

Abstract

What follows is an expanded version of my lectures at the NATO School on Equidistribution. I have tried to keep the informal style of the lectures. In particular, I have sometimes oversimplified matters in order to convey the spirit of an argument.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 299.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 379.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 379.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Balog, A. and Wooley, T. D. (2000) Sums of two squares in short intervals, Canad. J. Math. 52, 673–694.

    MATH  MathSciNet  Google Scholar 

  • Bogomolny, E. B. and Keating, J. P. (1996) Random matrix theory and the Riemann zeros. II. n-point correlations, Nonlinearity 9, 911–935.

    Article  MATH  MathSciNet  Google Scholar 

  • Chan, T. H. (2002) Pair correlation and distribution of prime numbers, Ph.D. thesis, University of Michigan.

    Google Scholar 

  • Chan, T. H. (2006) A note on primes in short intervals, Int. J. Number Theory 2, 105–110.

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  • Davenport, H. (2000) Multiplicative number theory, Vol. 74 of Grad. Texts in Math., New York, Springer.

    Google Scholar 

  • Feller, W. (1966) An introduction to probability theory and its applications, New York—London-Sydney, Wiley.

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Goldston, D. (2005) Notes on pair correlation of zeros and prime numbers, In Recent perspectives in random matrix theory and number theory, Vol. 322 of London Math. Soc. Lecture Notes Ser, Cambridge, Cambridge Univ. Press, pp. 79–110.

    Google Scholar 

  • Goldston, D. and Montgomery, H. L. (1987) On pair correlations of zeros and primes in short intervals, In Analytic number theory and Diophantine problems, Vol. 70 of Prog. Math., Stillwater, OK, 1984, pp. 183–203, Boston, Birkhäuser.

    Google Scholar 

  • Goldston, D., Pintz, J., and Yildirim, C. (2006) Primes in tuples. I, Ann. of Math. (2), to appear; preprint available at www.arxiv.org.

    Google Scholar 

  • Granville, A. (1995) Unexpected irregularities in the distribution of prime numbers, In Proceedings of the International Congress of Mathematicians. Vol. 1, 2, Zürich, 1994, pp. 388–399, Basel, Birkhäuser.

    Google Scholar 

  • Granville, A. and Martin, G. (2006) Prime number races, Amer. Math. Monthly 113, 1–33.

    Article  MathSciNet  MATH  Google Scholar 

  • Granville, A. and Soundararajan, K. (2006a) Sieving and the ErdÅ‘s—Kac theorem, this book.

    Google Scholar 

  • Granville, A. and Soundararajan, K. (2006b) An uncertainty principle for arithmetic sequences, Ann. of Math.(2), to appear; preprint available at www.arxiv.org.

    Google Scholar 

  • Hardy, G. H. and Littlewood, J. E. (1922) Some problems of Paritio Numerorum. III. On the expression of a number as a sum of primes, Acta Math. 44, 1–70.

    Article  MATH  MathSciNet  Google Scholar 

  • Heath-Brown, D. R. (1988) Differences between consecutive primes, Jahresber. Deutsch. Math.-Verein. 90, 71–89.

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  • Hooley, C. (1965) On the difference between consecutive numbers prime to n. II, Publ. Math. Debrecen 12, 39–49.

    MATH  MathSciNet  Google Scholar 

  • Hughes, C. P. and Rudnick, Z. (2004) On the distribution of lattice points in thin annuli, Int. Math. Res. Not. 2004, 637–658.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Monach, W. (1980) Numerical investigation of several problems in number theory, Ph.D. thesis, University of Michigan.

    Google Scholar 

  • Montgomery, H. L. (1973) The pair corelation of zeros of the zeta function, In Analytic Number Theory, Vol. 24 of Proc. Sympos. Pure Math., St. Louis Univ., 1972, pp. 181–193, Providence, RI, Amer. Math. Soc.

    Google Scholar 

  • Montgomery, H. L. and Soundararajan, K. (2002) Beyond pair correlation, In Paul ErdÅ‘s and his mathematics. I, Vol. 11 of Bolyai Soc. Math. Stud., Budapest, 1999, pp. 507–514, Budapest, János Bolyai Math. Soc.

    Google Scholar 

  • Montgomery, H. L. and Soundararajan, K. (2004) Primes in short intervals, Comm. Math. Phys. 252, 589–617.

    Article  MATH  MathSciNet  Google Scholar 

  • Montgomery, H. L. and Vaughan, R. C. (1986) On the distribution of reduced residues, Ann. of Math. (2) 123, 311–333.

    Article  MathSciNet  Google Scholar 

  • Rains, E. M. (1997) High powers of random elements of compact Lie groups, Probab. Theory Related Fields 107, 219–241.

    Article  MATH  MathSciNet  Google Scholar 

  • Rubinstein, M. and Sarnak, P. (1994) Chebyshev’s bias, Experimental Math. 3, 173–197.

    MATH  MathSciNet  Google Scholar 

  • Selberg, A. (1989) On the normal density of primes in short intervals, and the difference between consecutive primes, In Collected papers. Vol. I, Berlin, Springer, pp. 160–178.

    Google Scholar 

  • Soundararajan, K. (2006) Small gaps between prime numbers: the work of Goldston—Pintz—Yildirim, Bull. Amer. Math. Soc., to appear; preprint available at www.arxiv.org.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer

About this paper

Cite this paper

Soundararajan, K. (2007). THE DISTRIBUTION OF PRIME NUMBERS. In: Granville, A., Rudnick, Z. (eds) Equidistribution in Number Theory, An Introduction. NATO Science Series, vol 237. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-5404-4_4

Download citation

Publish with us

Policies and ethics