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Piatetski-Shapiro prime k-tuplets

Part of: Multiplicative number theory

Published online by Cambridge University Press:  26 February 2010

Glyn Harman
Glyn Harman
Affiliation:
School of Mathematics, Cardiff University of Wales, P.O. Box 926, Cardiff CF2 4YH.
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Extract

The question as to which “natural” sequences contain infinitely many primes is of considerable fascination to the number-theorist. One such “natura” sequence is [nc[ where [·] denotes integer part. Piatetski-Shapiro [10] showed that there are infinitely many primes in this sequence for 1 < c < 12/11, obtaining the expected asymptotic formula for the number of such primes. The exponent 12/11 has been increased gradually by a number of authors to the present record 45/38 held by Kumchev [9]. It is expected that there are infinitely many primes of the form [nc[ for all cεε[1, ∞)/ℤ. Deshouillers [3] showed that this is almost always true, in the sense of Lebesgue measure on [1, ∞). Balog [2] improved and generalized this result to show that, for almost all c > 1,

where

MSC classification

Secondary: 11N05: Distribution of primes
Type
Research Article
Information
Mathematika , Volume 45 , Issue 2 , December 1998 , pp. 351 - 357
Copyright
Copyright © University College London 1998

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References

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