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Primes in short intervals

Published online by Cambridge University Press:  01 September 1997

HONGZE LI
HONGZE LI
Affiliation:
Department of Mathematics, Shandong University, Jinan, Shandong, People's Republic of China
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Abstract

In 1982, Glyn Harman [2] proved that for almost all n, the interval [n, n+n(1/10)+ε] contains a prime number. By this we mean that the set of n[les ]N for which the interval does not contain a prime has measure o(N) as n→+∞. It follows from Huxley's work [6] that if θ>1/6 then there will almost always be asymptotically nθ(log n)−1 primes in the interval [n, n+nθ]. In 1983, Glyn Harman [3] pointed that for almost all n, the interval [n, n+n(1/12)+ε] contains a prime number, and meantime Heath-Brown gave the outline of this result in [5]. The exponent was reduced to 1/13 by Jia [10], 2/27 by Li [12] and 1/14 by Jia [11], and meantime N. Watt [16] got the same result. In this paper we shall prove the following result.

THEOREM. For almost all n, the interval

formula here

contains a prime number.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 122 , Issue 2 , September 1997 , pp. 193 - 205
Copyright
Cambridge Philosophical Society 1997

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Footnotes

This work is supported by the National Natural Science Foundations of China.