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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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Cited by 151 (26 self)
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Abstract. We prove that there are arbitrarily long arithmetic progressions of primes.
Obstructions to uniformity, and arithmetic patterns in the primes, preprint
"... Abstract. 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 iden ..."
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Cited by 5 (3 self)
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Abstract. 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. 1.
ON MANIN’S CONJECTURE FOR A FAMILY OF CHÂTELET SURFACES
"... Abstract. — The Manin conjecture is established for Châtelet surfaces over Q arising as minimal proper smooth models of the surface Y 2 + Z 2 = f(X) in A3 Q, where f ∈ Z[X] is a totally reducible polynomial of degree 3 without repeated roots. These surfaces do not satisfy weak approximation. Content ..."
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Cited by 3 (3 self)
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Abstract. — The Manin conjecture is established for Châtelet surfaces over Q arising as minimal proper smooth models of the surface Y 2 + Z 2 = f(X) in A3 Q, where f ∈ Z[X] is a totally reducible polynomial of degree 3 without repeated roots. These surfaces do not satisfy weak approximation. Contents 1. Introduction.................................................
Arithmetic progressions and the primes  El Escorial lectures
 Collectanea Mathematica (2006), Vol. Extra., 3788 (Proceedings of the 7th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial
"... Abstract. We describe some of the machinery behind recent progress in establishing infinitely many arithmetic progressions of length k in various sets of integers, in particular in arbitrary dense subsets of the integers, and in the primes. 1. ..."
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Cited by 3 (0 self)
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Abstract. We describe some of the machinery behind recent progress in establishing infinitely many arithmetic progressions of length k in various sets of integers, in particular in arbitrary dense subsets of the integers, and in the primes. 1.
SUMS OF ARITHMETIC FUNCTIONS OVER VALUES OF BINARY FORMS
, 2006
"... Abstract. Given a suitable arithmetic function h: N → R�0, and a binary form F ∈ Z[x1, x2], we investigate the average order of h as it ranges over the values taken by F. A general upper bound is obtained for this quantity, in which the dependence upon the coefficients of F is made completely explic ..."
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Abstract. Given a suitable arithmetic function h: N → R�0, and a binary form F ∈ Z[x1, x2], we investigate the average order of h as it ranges over the values taken by F. A general upper bound is obtained for this quantity, in which the dependence upon the coefficients of F is made completely explicit. 1.
Lecture 1 — Introduction to the
"... The most important analytic method for handling equidistribution questions about rational points on algebraic varieties is undoubtedly the HardyLittlewood circle method. There are a number of good texts available on the ..."
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The most important analytic method for handling equidistribution questions about rational points on algebraic varieties is undoubtedly the HardyLittlewood circle method. There are a number of good texts available on the
BINARY LINEAR FORMS AS SUMS OF TWO SQUARES
, 712
"... Abstract. We revisit recent work of HeathBrown on the average order of the quantity r(L1(x)) · · · r(L4(x)), for suitable binary linear forms L1,..., L4, as x = (x1, x2) ranges over quite general regions in Z 2. In addition to improving the error term in HeathBrown’s estimate we generalise his ..."
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Abstract. We revisit recent work of HeathBrown on the average order of the quantity r(L1(x)) · · · r(L4(x)), for suitable binary linear forms L1,..., L4, as x = (x1, x2) ranges over quite general regions in Z 2. In addition to improving the error term in HeathBrown’s estimate we generalise his result to cover a wider class of linear forms. 1.