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The primes contain arbitrarily long arithmetic progressions
 Ann. of Math
"... Abstract. We prove that there are arbitrarily long arithmetic progressions of primes. ..."
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Abstract. We prove that there are arbitrarily long arithmetic progressions of primes.
Linear equations in primes
 Annals of Mathematics
"... Abstract. Consider a system Ψ of nonconstant affinelinear forms ψ1,..., ψt: Z d → Z, no two of which are linearly dependent. Let N be a large integer, and let K ⊆ [−N, N] d be convex. A generalisation of a famous and difficult open conjecture of Hardy and Littlewood predicts an asymptotic, as N → ..."
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Abstract. Consider a system Ψ of nonconstant affinelinear forms ψ1,..., ψt: Z d → Z, no two of which are linearly dependent. Let N be a large integer, and let K ⊆ [−N, N] d be convex. A generalisation of a famous and difficult open conjecture of Hardy and Littlewood predicts an asymptotic, as N → ∞, for the number of integer points n ∈ Z d ∩ K for which the integers ψ1(n),..., ψt(n) are simultaneously prime. This implies many other wellknown conjectures, such as the twin prime conjecture and the (weak) Goldbach conjecture. It also allows one to count the number of solutions in a convex range to any simultaneous linear system of equations, in which all unknowns are required to be prime. In this paper we (conditionally) verify this asymptotic under the assumption that no two of the affinelinear forms ψ1,..., ψt are affinely related; this excludes the important “binary ” cases such as the twin prime or Goldbach conjectures, but does allow one to count “nondegenerate ” configurations such as arithmetic progressions. Our result assumes two families of conjectures, which we term the inverse Gowersnorm conjecture (GI(s)) and the Möbius and nilsequences conjecture (MN(s)), where s ∈ {1, 2,...} is
The GreenTao Theorem on arithmetic progressions in the primes: an ergodic point of view
, 2005
"... A longstanding and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. In an a ..."
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A longstanding and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. In an amazing fusion of methods from analytic number theory and ergodic theory, Ben Green and Terence Tao showed that for any positive integer k, there exist infinitely many arithmetic progressions of length k consisting only of prime numbers. This is an introduction to some of the ideas in the proof, concentrating on the connections to ergodic theory.
Quadratic uniformity of the Möbius function
, 2005
"... Abstract. This paper is a part of our programme to generalise the HardyLittlewood method to handle systems of linear questions in primes. This programme is laid out in our paper Linear equations in primes [14]. In particular, the results of this paper may be used, together with the machinery of [14 ..."
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Abstract. This paper is a part of our programme to generalise the HardyLittlewood method to handle systems of linear questions in primes. This programme is laid out in our paper Linear equations in primes [14]. In particular, the results of this paper may be used, together with the machinery of [14], to establish an asymptotic for the number of fourterm progressions p1 < p2 < p3 < p4 � N of primes, and more generally any problem counting prime points inside a “nondegenerate ” affine lattice of codimension at most 2. The main result of this paper is a proof of the Möbius and Nilsequences Conjecture for 1 and 2step nilsequences. This conjecture is introduced in [14] and amounts to showing that if G/Γ is an sstep nilmanifold, s � 2, if F: G/Γ → [−1, 1] is a Lipschitz function, and if Tg: G/Γ → G/Γ is the action of g ∈ G on G/Γ, then
What is good mathematics
, 2007
"... Abstract. Some personal thoughts and opinions on what “good quality mathematics” is, and whether one should try to define this term rigorously. As a case study, the story of Szemerédi’s theorem is presented. 1. The many aspects of mathematical quality We all agree that mathematicians should strive t ..."
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Abstract. Some personal thoughts and opinions on what “good quality mathematics” is, and whether one should try to define this term rigorously. As a case study, the story of Szemerédi’s theorem is presented. 1. The many aspects of mathematical quality We all agree that mathematicians should strive to produce good mathematics. But how does one define “good mathematics”, and should one even dare to try at all? Let us first consider the former question. Almost immediately one realises that there are many different types of mathematics which could be designated “good”. For instance, “good mathematics ” could refer (in no particular order) to (i) Good mathematical problemsolving (e.g. a major breakthrough on an important mathematical problem); (ii) Good mathematical technique (e.g. a masterful use of existing methods, or the development of new tools); (iii) Good mathematical theory (e.g. a conceptual framework or choice of notation which systematically unifies and generalises an existing body of results);
Obstructions to uniformity, and arithmetic patterns in the primes
, 2005
"... In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing in [26] that the primes contain arbitrarily long arithmetic progressions. One of the driving philosophies is to identify prec ..."
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Cited by 9 (5 self)
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In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing in [26] that the primes contain arbitrarily long arithmetic progressions. One of the driving philosophies is to identify precisely what the obstructions could be that prevent the primes (or any other set) from behaving “randomly”, and then either show that the obstructions do not actually occur, or else convert the obstructions into usable structural information on the primes.
DIFFERENCE SETS AND POLYNOMIALS OF PRIME VARIABLES
, 709
"... Abstract. Let ψ(x) be a polynomial with rational coefficients. Suppose that ψ has the positive leading coefficient and zero constant term. Let A be a set of positive integers with the positive upper density. Then there exist x, y ∈ A and a prime p such that x − y = ψ(p − 1). Furthermore, if P be a s ..."
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Cited by 7 (2 self)
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Abstract. Let ψ(x) be a polynomial with rational coefficients. Suppose that ψ has the positive leading coefficient and zero constant term. Let A be a set of positive integers with the positive upper density. Then there exist x, y ∈ A and a prime p such that x − y = ψ(p − 1). Furthermore, if P be a set of primes with the positive relative upper density, then there exist x, y ∈ P and a prime p such that x − y = ψ(p − 1). For a set A of positive integers, define 1.
A DENSITY VERSION OF VINOGRADOV’S THREE PRIMES THEOREM
, 2008
"... Abstract. Let P denote the set of all primes. Suppose that P1, P2, P3 are three subsets of P with d P (P1) + d P (P2) + d P (P3)> 2, where d P (Pi) is the lower density of Pi relative to P. We prove that for every sufficiently large odd integer n, there exist pi ∈ Pi such that n = p1 + p2 + p3. 1 ..."
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Abstract. Let P denote the set of all primes. Suppose that P1, P2, P3 are three subsets of P with d P (P1) + d P (P2) + d P (P3)> 2, where d P (Pi) is the lower density of Pi relative to P. We prove that for every sufficiently large odd integer n, there exist pi ∈ Pi such that n = p1 + p2 + p3. 1.
On sums of sets of primes with positive relative density
 J. Lond. Math. Soc
, 2011
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Long arithmetic progressions of primes
 Mathematics Proceedings
"... Abstract. This is an article for a general mathematical audience on the author’s work, joint with Terence Tao, establishing that there are arbitrarily long arithmetic progressions of primes. 1. introduction and history This is a description of recent work of the author and Terence Tao [11] on primes ..."
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Abstract. This is an article for a general mathematical audience on the author’s work, joint with Terence Tao, establishing that there are arbitrarily long arithmetic progressions of primes. 1. introduction and history This is a description of recent work of the author and Terence Tao [11] on primes in arithmetic progression. It is based on seminars given for a general mathematical