Subtleties in the Distribution of Primes

  • Hans Riesel
Part of the Modern Birkhäuser Classics book series (MBC)


There are only very few proved results concerning the distribution of primes in short intervals. The prime number theorem tells us that the average density of primes around x is approximately 1/In x. This means that if we consider an interval of length Δx about x and choose any integer t in this interval, then the probability of t being a prime will approach 1/ In x as x → ∞, if Δx is small compared to x. This implies that the primes tend to thin out as x grows larger; an implication that becomes obvious when considering that the condition for a randomly picked integer x to be composite is that it has some prime factor \( \leqslant \sqrt x\) and that there are more prime factors \( \leqslant \sqrt x\) to choose from when x is larger.


Number Series Residue Class White Ball Infinite Product Consecutive Integer 
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  1. 1.
    Daniel Shanks and John W. Wrench, Jr., “Brun’s Constant,” Math. Comp. 28 (1974) pp. 293–299.MathSciNetGoogle Scholar
  2. 1′.
    B. K. Parady, Joel F. Smith, and Sergio E. Zarantonello, “Largest Known Twin Primes,” Math. Comp. 55 (1990) pp. 381–382.MathSciNetMATHCrossRefGoogle Scholar
  3. 2.
    G. H. Hardy and J. E. Littlewood, “Some Problems of ‘Partitio Numerorum’ III: On the Expression of a Number as a Sum of Primes,” Acta Math. 44 (1922) pp. 1–70 = G. H. Hardy, Coll. Papers, vol. 1, pp. 561–630.MathSciNetCrossRefGoogle Scholar
  4. 3.
    F. J. Gruenberger and G. Armerding, Statistics on the First Six Million Prime Numbers, Reviewed in Math. Comp. 19 (1965) pp. 503–505.Google Scholar
  5. 4.
    Hans Riesel and Robert. C. Vaughan, “On Sums of Primes,” Arkiv fr Mat. 21 (1983) pp. 45–74.MathSciNetMATHCrossRefGoogle Scholar
  6. 5.
    G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth edition, Oxford, 1979, p. 351.MATHGoogle Scholar
  7. 6.
    Richard P. Brent, “The Distribution of Small Gaps Between Successive Primes,” Math. Comp. 28 (1974) pp. 315–324.MathSciNetMATHCrossRefGoogle Scholar
  8. 7.
    Ian Richards, “On the Incompatibility of Two Conjectures Concerning Primes,” Bull. Am. Math. Soc. 80 (1974) pp. 419–438.MathSciNetMATHCrossRefGoogle Scholar
  9. 8.
    Thomas Vehka, “Explicit Construction of an Admissible Set for the Conjecture that Sometimes π(x + y) > π(y) + π(x)” Notices Am. Math. Soc. 26 (1979) p. A-453.Google Scholar
  10. 9.
    Hans Riesel, “Primes Forming Arithmetic Series and Clusters of Large Primes,” Nordisk Tidskrift för Informationsbehandling (BIT) 10 (1970) pp. 333–342.MathSciNetMATHGoogle Scholar
  11. 10.
    J. E. Littlewood, “Sur la Distribution des Nombres Premiers,” Comptes Rendus 158 (1914) pp. 1869–1872.MATHGoogle Scholar
  12. 11.
    Carter Bays and Richard H. Hudson, “On the Fluctuations of Littlewood for Primes of the Form 4n ± 1,” Math. Comp. 32 (1978) pp. 281–286.MathSciNetMATHGoogle Scholar
  13. 12.
    William Feller, An Introduction to Probability Theory and its Applications, vol. I, Second edition, Wiley, New York, 1957, pp. 73–87.MATHGoogle Scholar
  14. 13.
    Carter Bays and Richard H. Hudson, “Details of the First Region of Integers x with π3,2 x < π3,1 xMath. Comp. 32 (1978) pp. 571–576.MathSciNetMATHGoogle Scholar
  15. 14.
    Daniel Shanks, “On Maximal Gaps Between Successive Primes,” Math. Comp. 18 (1964) pp. 646–651.MathSciNetCrossRefGoogle Scholar
  16. 15.
    L. J. Lander and T. R. Parkin, “On the First Appearance of Prime Differences,” Math. Comp. 21 (1967) pp. 483–488.MathSciNetMATHCrossRefGoogle Scholar
  17. 16.
    Richard P. Brent, “The First Occurrence of Large Gaps Between Successive Primes,” Math. Comp. 27 (1973) pp. 959–963.MathSciNetMATHCrossRefGoogle Scholar
  18. 17.
    Richard P. Brent, “The First Occurrence of Certain Large Prime Gaps,” Math. Comp. 35 (1980) pp. 1435–1436.MathSciNetMATHCrossRefGoogle Scholar
  19. 18.
    Jeff Young and Aaron Potier, “First Occurrence Prime Gaps, Math. Comp. 52 (1989) pp. 221–224.MathSciNetMATHCrossRefGoogle Scholar
  20. 19.
    Harald Cramér, “On the Order of Magnitude of the Difference Between Consecutive Prime Numbers,” Acta Arith. 2 (1936) pp. 23–46.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsThe Royal Institute of TechnologyStockholmSweden

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