Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-19T05:48:04.174Z Has data issue: false hasContentIssue false
  • English
  • Français

The Difference between Consecutive Prime Numbers V

Published online by Cambridge University Press:  20 January 2009

R. A. Rankin
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let pn denote the nth prime and let ε be any positive number. In 1938 (3) Ishowed that, for an infinity of values of n,

where, for k≧1, logk+1x = log (logk x) and log1x = log x. In a recent paper (4) Schönhage has shown that the constant ⅓ may be replaced by the larger number ½eγ, where γ is Euler's constant; this is achieved by means of a more efficient selection of the prime moduli used. Schönhage uses an estimate of mine for the number B1 of positive integers n≦u that consist entirely of prime factors p≦y, where

Herer x is large and α and δ are positive constants to be chosen suitably.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 13 , Issue 4 , December 1963 , pp. 331 - 332
Copyright
Copyright © Edinburgh Mathematical Society 1963

References

REFERENCES

De Bruijn, N. G.On the number of positive integers ≦x and free of prime factors >y, Nederl. Akad. Wetensch. Proc., (A), 54 (1951), 50–60.CrossRefGoogle Scholar
De Bruijn, N. G.The asymptotic behaviour of a function occurring in the theory of primes, J. Indian Math. Soc., 15 (1951), 25–32Google Scholar
Rankin, R. A.The difference between consecutive prime numbers, J. London Math. Soc. 13 (1938), 242–247.CrossRefGoogle Scholar
Schönhage, A.Eine Bemerkung zur Konstruktion grosser Primzahllücken, Arch. Math., 14 (1963), 29–30.CrossRefGoogle Scholar