THE DISTRIBUTION OF PRIME NUMBERS

  • K. Soundararajan
Part of the NATO Science Series book 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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Balog, A. and Wooley, T. D. (2000) Sums of two squares in short intervals, Canad. J. Math. 52, 673–694.MATHMathSciNetGoogle Scholar
  2. Bogomolny, E. B. and Keating, J. P. (1996) Random matrix theory and the Riemann zeros. II. n-point correlations, Nonlinearity 9, 911–935.MATHCrossRefMathSciNetGoogle Scholar
  3. Chan, T. H. (2002) Pair correlation and distribution of prime numbers, Ph.D. thesis, University of Michigan.Google Scholar
  4. Chan, T. H. (2006) A note on primes in short intervals, Int. J. Number Theory 2, 105–110.MATHCrossRefMathSciNetGoogle Scholar
  5. Cramér, H. (1936) On the order of magnitude of the difference between consecutive prime numbers, Acta Arith. 2, 23–46.MATHGoogle Scholar
  6. Davenport, H. (2000) Multiplicative number theory, Vol. 74 of Grad. Texts in Math., New York, Springer.Google Scholar
  7. Feller, W. (1966) An introduction to probability theory and its applications, New York—London-Sydney, Wiley.MATHGoogle Scholar
  8. Friedlander, J. and Granville, A. (1989) Limitations to the equi-distribution of primes. I, Annals of Math. (2) 129, 363–382.CrossRefMathSciNetGoogle Scholar
  9. Gallagher, P. X. (1976) On the distribution of primes in short intervals, Mathematika 23, 4–9.MathSciNetCrossRefMATHGoogle Scholar
  10. 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
  11. 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
  12. 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
  13. 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
  14. Granville, A. and Martin, G. (2006) Prime number races, Amer. Math. Monthly 113, 1–33.MathSciNetMATHCrossRefGoogle Scholar
  15. Granville, A. and Soundararajan, K. (2006a) Sieving and the Erdős—Kac theorem, this book.Google Scholar
  16. 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
  17. 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.CrossRefMATHMathSciNetGoogle Scholar
  18. Heath-Brown, D. R. (1988) Differences between consecutive primes, Jahresber. Deutsch. Math.-Verein. 90, 71–89.MATHMathSciNetGoogle Scholar
  19. Hildebrand, A. and Maier, H. (1989) Irregularities in the distribution of primes in short intervals, J. Reine Angew. Math. 397, 162–193.MATHMathSciNetGoogle Scholar
  20. Hooley, C. (1965) On the difference between consecutive numbers prime to n. II, Publ. Math. Debrecen 12, 39–49.MATHMathSciNetGoogle Scholar
  21. Hughes, C. P. and Rudnick, Z. (2004) On the distribution of lattice points in thin annuli, Int. Math. Res. Not. 2004, 637–658.MATHCrossRefMathSciNetGoogle Scholar
  22. Maier, H. (1985) Primes in short intervals, Michigan Math. J. 32, 221–225.MATHCrossRefMathSciNetGoogle Scholar
  23. Monach, W. (1980) Numerical investigation of several problems in number theory, Ph.D. thesis, University of Michigan.Google Scholar
  24. 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
  25. Montgomery, H. L. and Soundararajan, K. (2002) Beyond pair correlation, In Paul Erdős and his mathematics. I, Vol. 11 of Bolyai Soc. Math. Stud., Budapest, 1999, pp. 507–514, Budapest, János Bolyai Math. Soc.Google Scholar
  26. Montgomery, H. L. and Soundararajan, K. (2004) Primes in short intervals, Comm. Math. Phys. 252, 589–617.MATHCrossRefMathSciNetGoogle Scholar
  27. Montgomery, H. L. and Vaughan, R. C. (1986) On the distribution of reduced residues, Ann. of Math. (2) 123, 311–333.CrossRefMathSciNetGoogle Scholar
  28. Rains, E. M. (1997) High powers of random elements of compact Lie groups, Probab. Theory Related Fields 107, 219–241.MATHCrossRefMathSciNetGoogle Scholar
  29. Rubinstein, M. and Sarnak, P. (1994) Chebyshev’s bias, Experimental Math. 3, 173–197.MATHMathSciNetGoogle Scholar
  30. 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
  31. 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

Copyright information

© Springer 2007

Authors and Affiliations

  • K. Soundararajan
    • 1
  1. 1.University of Michigan

Personalised recommendations