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Subtleties in the Distribution of Primes

  • Hans Riesel
Part of the Progress in Mathematics book series (PM, volume 57)

Abstract

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/ln 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/ln 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 \( \le \sqrt x \), and that there are more prime factors \( \le \sqrt x \) to choose from when x is larger.

Keywords

Number Series Residue Class Successive Prime White Ball Infinite Product 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Bibliography

  1. 1.
    Daniel Shanks and John W. Wrench Jr., “Brun’s Constant,” Math. Comp. 28 (1974) pp. 293–299.MathSciNetzbMATHGoogle Scholar
  2. 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
  3. 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
  4. 4.
    Hans Riesel and Robert. C. Vaughan, “On Sums of Primes,” Arkiv für Mat. 21 (1983) pp. 45–74.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth edition, Oxford, 1979, p. 351.zbMATHGoogle Scholar
  6. 6.
    Richard P. Brent, “The Distribution of Small Gaps Between Successive Primes,” Math. Comp. 28 (1974) pp. 315–324.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Ian Richards, “On the Incompatibility of Two Conjectures Concerning Primes,” Bull. Amer. Math. Soc. 80 (1974) pp. 419–438.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 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
  9. 9.
    Hans Riesel, “Primes Forming Arithmetic Series and Clusters of Large Primes,” Nordisk Tidskr. för Informationsbehandling (BIT) 10 (1970) pp. 333–342.MathSciNetzbMATHGoogle Scholar
  10. 10.
    J. E. Littlewood, “Sur la Distribution des Nombres Premiers,” Comptes Rendus 158 (1914) pp. 1869–1872.zbMATHGoogle Scholar
  11. 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.MathSciNetzbMATHGoogle Scholar
  12. 12.
    William Feller, An Introduction to Probability Theory and its Applications, vol. I, Second edition, Wiley, New York, 1957, pp. 73–87.zbMATHGoogle Scholar
  13. 13.
    Carter Bays and Richard H. Hudson, “Details of the First Region of Integers x with π 3,2 (x) < π 3,1 (x),” Math. Comp.32 (1978) pp. 571–576.MathSciNetzbMATHGoogle Scholar
  14. 14.
    Daniel Shanks, “On Maximal Gaps Between Successive Primes,” Math. Comp. 18 (1964) pp. 646–651.MathSciNetCrossRefGoogle Scholar
  15. 15.
    L. J. Lander and T. R. Parkin, “On the First Appearance of Prime Differences,” Math. Comp. 21 (1967) pp. 483–488.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Richard P. Brent, “The First Occurrence of Large Gaps Between Successive Primes,” Math. Comp. 27 (1973) pp. 959–963.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Richard P. Brent, “The First Occurrence of Certain Large Prime Gaps,” Math. Comp. 35 (1980) pp. 1435–1436.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    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 New York 1985

Authors and Affiliations

  • Hans Riesel
    • 1
  1. 1.VällingbySweden

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