Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-25T02:48:28.013Z Has data issue: false hasContentIssue false
  • English
  • Français

Sifting short intervals

Published online by Cambridge University Press:  24 October 2008

John B. Friedlander
John B. Friedlander
Affiliation:
University of Toronto.
Get access Rights & Permissions [Opens in a new window]

Extract

Let φ (X) denote a function satisfying limx→∞, φ(X) = ∞. Let ℬ be a sequence of positive integers and let T(X) = T(X, ℬ, φ) be the set of real numbers y, 1 ≤ y ≤ X, such that the interval (y, y + φ(y)] contains no member of ℬ.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 91 , Issue 1 , January 1982 , pp. 9 - 15
Copyright
Copyright © Cambridge Philosophical Society 1982

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Heath-Brown, D. R.Almost-primes in arithmetic progressions and short intervals. Math. Proc. Cambridge Philos. Soc. 83 (1978), 357–375.CrossRefGoogle Scholar
(2)Heath-Brown, D. R. Gaps between primes, and the pair correlation of zeros of the zetafunction. Acta Arith. to appear.Google Scholar
(3)Hooley, C.On the intervals between numbers that are sums of two squares. Acta Math. 127 (1971), 279–297.CrossRefGoogle Scholar
(4)Huxley, M. N.The distribution of prime numbers. Oxford, London, 1972.Google Scholar
(5)Ingham, A. E.Some asymptotic formulae in the theory of numbers. J.L.M.S. 2 (1927), 202–208.Google Scholar
(6)Iwaniec, H.Rosser's sieve. Acta Arith. 36 (1980), 171–202.CrossRefGoogle Scholar
(7)Motohashi, Y.On almost primes in arithmetic progressions. J. Math. Soc. Japan 28 (1976), 363–383.Google Scholar
(8)Motohashi, Y.A note on almost-primes in short intervals. Proc. Japan Acad. Ser. A, 55 (1979), 225–226.Google Scholar
(9)Nicolas, J. L.Répartition des nombres largements composés. Acta Arith. 34 (1979), 379–390.CrossRefGoogle Scholar
(10)Selberg, A.On the normal density of primes in small intervals and the difference between consecutive primes. Arch Math. Naturvid. 47 (1943), 87–105.Google Scholar
(11)Wolke, D.Fast-Primzahlen in kurzen Intervallen. Math. Ann. 244 (1979), 233–242.CrossRefGoogle Scholar