Abstract. We prove that there are arbitrarily long arithmetic progressions of primes.

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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.

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.

The most important analytic method for handling equidistribution questions about rational points on algebraic varieties is undoubtedly the Hardy-Littlewood circle method. There are a number of good texts available on the

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.

Abstract. We revisit recent work of Heath-Brown 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 Heath-Brown’s estimate we generalise his result to cover a wider class of linear forms. 1.