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The Difference between Consecutive Prime Numbers V

Published online by Cambridge University Press:  20 January 2009

R. A. Rankin
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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
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Schönhage, A.Eine Bemerkung zur Konstruktion grosser Primzahllücken, Arch. Math., 14 (1963), 29–30.CrossRefGoogle Scholar