A smoothed GPY sieve

@article{Motohashi2008ASG,
  title={A smoothed GPY sieve},
  author={Yoichi Motohashi and Janos Pintz},
  journal={Bulletin of the London Mathematical Society},
  year={2008},
  volume={40}
}
  • Y. Motohashi, J. Pintz
  • Published 27 February 2006
  • Mathematics
  • Bulletin of the London Mathematical Society
Combining the arguments developed in the works of D. A. Goldston, S. W. Graham, J. Pintz, and C. Y. Yildirim [Preprint, 2005, arXiv: math.NT/506067] and Y. Motohashi [Number theory in progress – A. Schinzel Festschrift (de Gruyter, 1999) 1053–1064] we introduce a smoothing device to the sieve procedure of Goldston, Pintz, and Yildirim (see [Proc. Japan Acad. 82A (2006) 61–65] for its simplified version). Our assertions embodied in Lemmas 3 and 4 of this article imply that a natural extension of… 
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References

SHOWING 1-10 OF 20 REFERENCES
Small gaps between primes exist
In the recent preprint (3), Goldston, Pintz, and Yoldorom established, among other things, (0) liminf n→∞ pn+1 − pn log pn = 0, with pn the nth prime. In the present article, which is essentially…
Sieves in Number Theory
This book surveys the current state of the "small" sieve methods developed by Brun, Selberg and later workers.. A self-contained treatment is given to topics that are of central importance in the…
SMALL GAPS BETWEEN PRIMES II ( PRELIMINARY )
We examine an idea for approximating prime tuples. 1. Statement of results (Preliminary) In the present work we will prove the following result. Let pn denote the nth prime. Then (1.1) lim inf n→∞…
An Overview of the Sieve Method and its History
This is a revised version of NT0505521, a translation of our Japanese expository article that was published under the title `{\it An overview of sieve methods}' in the second issue of the 52nd volume…
Primes in arithmetic progressions to large moduli
Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
On the remainder term in the Selberg sieve
Lectures on sieve methods and prime number theory
E-mail: ymoto@math.cst.nihon-u.ac.jp János Pintz Rényi Mathematical Institute of the Hungarian Academy of Sciences, H-1364 Budapest
  • E-mail: ymoto@math.cst.nihon-u.ac.jp János Pintz Rényi Mathematical Institute of the Hungarian Academy of Sciences, H-1364 Budapest
Small gaps between primes or products of two primes
    Small gaps between primes or products of two primes
    • Small gaps between primes or products of two primes
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