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Bulletin of the London Mathematical Society Advance Access originally published online on April 12, 2008
Bulletin of the London Mathematical Society 2008 40(2):298-310; doi:10.1112/blms/bdn023
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© 2008 London Mathematical Society

A smoothed GPY sieve

Yoichi Motohashi

Department of Mathematics
Nihon University
Surugadai
Tokyo 101-8308
Japan
ymoto@math.cst.nihon-u.ac.jp

János Pintz

Rényi Mathematical Institute of the Hungarian Academy of Sciences
Reáltanoda u. 13–15
H-1053 Budapest
Hungary

Received 24 April 2007. Revision received 10 December 2007.

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 a prime number theorem of E. Bombieri, J. B. Friedlander, and H. Iwaniec [Theorem 8 in Acta Math. 156 (1986) 203–251] should give rise infinitely often to bounded differences between primes, that is, a weaker form of the twin prime conjecture.

2000 Mathematics Subject Classification 11N05 (primary), 11P32 (secondary).

The first author was supported by Grants-in-Aid for Scientific Research (C) 15540047, and the second author by OTKA grants nos T38396, T43623 and T49693 [GenBank] and the Balaton program.


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