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## Primes in tuples I

- D. Goldston, J. Pintz, C. Yildirim
- Mathematics
- 10 August 2005

We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the… Expand

## Small gaps between products of two primes

- D. Goldston, S. Graham, J. Pintz, C. Yildirim
- Mathematics
- 21 September 2006

Let qn denote the nth number that is a product of exactly two distinct primes. We prove that qn+1 − qn ⩽ 6 infinitely often. This sharpens an earlier result of the authors, which had 26 in place of… Expand

## Higher correlations of divisor sums related to primes II: variations of the error term in the prime number theorem

- D. Goldston, C. Yildirim
- Mathematics
- 18 December 2004

We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We… Expand

## Higher correlations of divisor sums related to primes III: small gaps between primes

- D. Goldston, C. Yildirim
- Mathematics
- 1 November 2007

We use divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that, for any η > 0, a… Expand

## Small gaps between primes exist

- D. Goldston, Y. Motohashi, J. Pintz, C. Yildirim
- Mathematics
- 14 May 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… Expand

## Primes in tuples IV: Density of small gaps between consecutive primes

- D. Goldston, J. Pintz, C. Yildirim
- Mathematics
- 30 March 2011

α e−t dt as x→∞, for any two fixed real numbers β > α ≥ 0. Gallagher’s calculation [Ga] shows that this conjecture can be deduced from the Hardy–Littlewood prime k-tuples conjecture (see [S2]). Hence… Expand

## Small gaps between almost primes, the parity problem, and some conjectures of Erdős on consecutive integers II

- D. Goldston, S. Graham, Apoorva Panidapu, J. Pintz, Jordan Schettler, C. Yildirim
- MathematicsJournal of Number Theory
- 18 March 2008

In a previous paper, the authors proved that in any system of three linear forms satisfying obvious necessary local conditions, there are at least two forms that infinitely often assume E 2 -values;… Expand

## Positive Proportion of Small Gaps Between Consecutive Primes

- D. Goldston, J. Pintz, C. Yildirim
- Mathematics
- 21 March 2011

We prove that a positive proportion of the gaps between consecutive primes are short gaps of length less than any fixed fraction of the average spacing between primes.

## Primes in Short Segments of Arithmetic Progressions

- D. Goldston, C. Yildirim
- MathematicsCanadian Journal of Mathematics
- 1 June 1998

Abstract Consider the variance for the number of primes that are both in the interval $\left[ y,\,y\,+\,h \right]$ for $y\,\in \,[x,\,2x]$ and in an arithmetic progression of modulus $q$ . We study… Expand

## The path to recent progress on small gaps between primes

- D. Goldston, J. Pintz, C. Yildirim
- Economics
- 19 December 2005

We present the development of ideas which led to our recent find- ings about the existence of small gaps between primes.

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