# 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}
}
• 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…
Close encounters among the primes
• Mathematics
• 2013
This paper was written, apart from one technical correction, in July and August of 2013. The, then very recent, breakthrough of Y. Zhang [18] had revived in us an intention to produce a second
N T ] 1 3 Ja n 20 14 CLOSE ENCOUNTERS AMONG THE PRIMES
Abstract. This paper was written, apart from one technical correction, in July and August of 2013. The, then very recent, breakthrough of Y. Zhang [Zha13] had revived in us an intention to produce a
Primes in intervals of bounded length
In April 2013, Yitang Zhang proved the existence of a finite bound B such that there are infinitely many pairs of distinct primes which differ by no more than B. This is a massive breakthrough, makes
New equidistribution estimates of Zhang type
In May 2013, Y. Zhang [52] proved the existence of infinitely many pairs of primes with bounded gaps. In particular, he showed that there exists at least one h ě 2 such that the set tp prime | p h
Variants of the Selberg sieve, and bounded intervals containing many primes
For any m≥1, let Hm denote the quantity liminfn→∞(pn+m−pn). A celebrated recent result of Zhang showed the finiteness of H1, with the explicit bound H1≤70,000,000. This was then improved by us (the
BOUNDED GAPS BETWEEN PRIMES
Recently, Yitang Zhang proved the existence of a finite bound B such that there are infinitely many pairs pn, pn 1 of consecutive primes for which pn 1 pn ¤ B. This can be seen as a massive
ON ZHANG ’ S PRIME GAPS PAPER
This result is deduced from the following result. Call an admissible set to be a finite set of integers H which avoids at least one residue class modulo p for each prime p. For any natural number k0,
Diagonal cubic forms and the large sieve
Let F (x) be a diagonal integer-coefficient cubic form in m ∈ {4, 5, 6} variables. Excluding rational lines if m = 4, we bound the number of integral solutions x ∈ [−X,X] to F (x) = 0 by OF, (X
On the Ratio of Consecutive Gaps Between Primes
In the present work we prove a common generalization of Maynard–Tao’s recent result about consecutive bounded gaps between primes and of the Erdős–Rankin bound about large gaps between consecutive
Primes in arithmetic progressions with friable indices
• Mathematics
Science China Mathematics
• 2019
We consider the number π(x, y; q, a) of primes p ⩽ x such that p ≡ a (mod q ) and ( p − a )/ q is free of prime factors greater than y. Assuming a suitable form of Elliott-Halberstam conjecture, it

## References

SHOWING 1-10 OF 21 REFERENCES
Small gaps between primes exist
• Mathematics
• 2005
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
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
• Mathematics
• 1986
Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
SMALL GAPS BETWEEN PRIMES II ( PRELIMINARY )
• Mathematics
• 2005
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→∞
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