Communications in Mathematical Physics

, Volume 252, Issue 1–3, pp 589–617 | Cite as

Primes in Short Intervals

  • Hugh L. MontgomeryEmail author
  • K. Soundararajan


Contrary to what would be predicted on the basis of Cramér’s model concerning the distribution of prime numbers, we develop evidence that the distribution of ψ(x+H)−ψ(x), for 0≤xN, is approximately normal with mean ∼H and variance ∼H log N/H, when N δ HN1− δ .


Neural Network Statistical Physic Complex System Nonlinear Dynamics Short Interval 
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  1. 1.
    Bogomolny, E.B., Keating, J.P.: Random matrix theory and the Riemann zeros. II. n-point correlations. Nonlinearity 9, 911–935 (1996)zbMATHGoogle Scholar
  2. 2.
    Brent, R.P.: Irregularities in the distribution of primes and twin primes. Math. Comp. 29, 43–56 (1975); Correction 30, 198 (1976)zbMATHGoogle Scholar
  3. 3.
    Chan, T.H.: Pair correlation and distribution of prime numbers. PhD dissertation University of Michigan Ann Arbor, 2002 vi+101Google Scholar
  4. 4.
    Cramér, H.: On the order of magnitude of the difference between consecutive prime numbers. Acta Arith. 2, 23–46 (1936)Google Scholar
  5. 5.
    Forrester, P.J., Odlyzko, A.M.: Gaussian unitary ensemble eigenvalues and Riemann ζ function zeros: a nonlinear equation for a new statistic. Phys. Rev. E (3) 54, R4493–R4495 (1996)Google Scholar
  6. 6.
    Gallagher, P.X.: On the distribution of primes in short intervals. Mathematika 23 4–9, 1976; Corrigendum, 28, 86 (1981)Google Scholar
  7. 7.
    Goldston, D.A.: Linnik’s theorem on Goldbach numbers in short intervals. Glasgow Math. J. 32, 285–297 (1990)zbMATHGoogle Scholar
  8. 8.
    Goldston, D.A., Montgomery, H.L.: On pair correlations of zeros and primes in short intervals. In: A. C. Adolphson, J. B. Conrey, A. Ghosh, R. I. Yager, (eds.), Analytic Number Theory and Diophantine Problems, Stillwater, OK, July 1984, Prog. Math. 70, Boston: Birkäuser, 1987, pp. 183–203Google Scholar
  9. 9.
    Granville, A., Soundararajan, K.: An uncertainty principle for arithmetic sequences., 2004
  10. 10.
    Hardy, G.H., Littlewood, J.E.: Some problems of Partitio Numerorum (III): On the expression of a number as a sum of primes. Acta Math. 44, 1–70 (1922)zbMATHGoogle Scholar
  11. 11.
    Hausman, M., Shapiro, H.N.: On the mean square distribution of primitive roots of unity. Commun. Pure App. Math. 26, 539–547 (1973)zbMATHGoogle Scholar
  12. 12.
    Maier, H.: Primes in short intervals. Michigan Math. J. 32, 221–225 (1985)CrossRefzbMATHGoogle Scholar
  13. 13.
    Montgomery, H.L.: The pair correlation of zeros of the zeta function. In: Analytic Number Theory (St. Louis Univ., 1972) Proc. Sympos. Pure Math. 24, Providence, RI: Am. Math. Soc. 1973, pp. 181–193Google Scholar
  14. 14.
    Montgomery, H.L.: Ten lectures on the interface between analytic number theory and harmonic analysis. CBMS 84, Providence, RI: Am. Math. Soc. 1994, xii+220Google Scholar
  15. 15.
    Montgomery, H.L., Soundararajan, K.: Beyond pair correlation. In: Paul Erdős and his Mathematics. I Math. Studies 11, Budapest: Bolyai Society 2002, pp. 507–514Google Scholar
  16. 16.
    Montgomery, H.L., Vaughan, R.C.: On the distribution of reduced residues. Ann. Math. 123, 311–333 (1986)zbMATHGoogle Scholar
  17. 17.
    Montgomery, H.L., Vaughan, R.C.: A basic inequality. In: Congress in Number Theory (Zarautz, 1984), Bilbao: Universidad del Paí s Vasco 1989, pp. 163–175Google Scholar
  18. 18.
    Odlyzko, A.M.: On the distribution of spacings between zeros of the zeta function. Math. Comp. 48, 273–308 (1987)zbMATHGoogle Scholar
  19. 19.
    Rains, E.M.: High powers of random elements of compact Lie groups. Probab. Theory Related Fields 107, 219–241 (1997)CrossRefzbMATHGoogle Scholar
  20. 20.
    Selberg, A.: On the normal density of primes in short intervals, and the difference between consecutive primes. Collected papers (Volume I), Berlin-Heidelberg-New York: Springer, 1989, pp. 160–178Google Scholar

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© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA

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