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Linear equations in primes. (English) Zbl 1242.11071
Let \(\psi_1,\ldots,\psi_t:\mathbb{Z}^d\rightarrow\mathbb{Z}\) be non-constant affine linear forms, and let \(K\) be a convex sub-region of \([-N,N]^d\). Then there is a natural conjecture, of Hardy–Littlewood type, for the number of \(\mathbf{n}\in K\cap\mathbb{Z}^d\) for which the values \(\psi_i(\mathbf{n})\) are simultaneously prime. This paper gives a conditional proof of the conjecture, under the assumption that no two of the functions \(\psi_i\) have homogeneous parts which are parallel. This condition rules out the pair of functions \(n,n+2\), which would have handled the twin prime conjecture. None the less the paper represents a very substantial step forward in our understanding of such problems. One particular case of interest is that in which the forms are \(n_1,n_1+n_2,n_2+2n_2,\ldots,n_1+(t-1)n_2\), for which one obtains an asymptotic formula for the number of \(t\)-term arithmetic progressions of primes in appropriate regions. Previously [Ann. Math. (2) 167, No. 2, 481–547 (2008; Zbl 1191.11025)] the authors had proved that there are infinitely many \(t\)-term arithmetic progressions of primes, but the method had not produced an asymptotic formula. In general the method of the present paper gives a conditional proof that \[ \sum_{\mathbf{n}\in K\cap\mathbb{Z}^d}\Lambda(\psi_1(\mathbf{n}))\ldots \Lambda(\psi_t(\mathbf{n}))=c\text{Vol}(K)+o(N^d) \] as \(N\rightarrow\infty\), where the constant \(c\) is the usual product of local densities.
As mentioned, the proof is a conditional one, subject to two hypotheses, namely that the “inverse Gowers norm conjecture GI(s)” and the “Möbius and nilsequences conjecture MN(s)” each hold for every integer \(s\geq 2\). The conjectures GI(2) and MN(2) had previously been proved by the authors [Proc. Edinb. Math. Soc., II. Ser. 51, No. 1, 73–153 (2008; Zbl 1202.11013)], and [Ann. Inst. Fourier 58, No. 6, 1863–1935 (2008; Zbl 1160.11017)], and these enable systems of linear forms of “complexity” at most 2 to be handled unconditionally. In particular the paper gives an unconditional proof for the case of sets of 4 primes in arithmetic progression.
Indeed, in subsequent work by the authors, together with Ziegler, the conjectures GI(s) have been established for all \(s\); and similarly the authors have proved MN(s) in all cases. Thus the results of the present paper are now known unconditionally.
The conjectures GI(s) and MN(s) are somewhat technical to describe. Very roughly, the inverse Gowers norm conjecture describes functions for which the Gowers norm \(U^{s+1}\) is large, saying this can only happen if the function correlates with an \(s\)-step nilsequence. The Möbius and nilsequences conjecture states roughly that the Möbius function \(\mu(n)\) does not correlate with \(s\)-step nilsequences. Thus one can think of MN(1) as saying that \(\sum_{n\leq N}\mu(n)e(\theta n)=o(N)\) for every real \(\theta\).
The paper builds on the authors’ earlier work on arithmetic progressions of primes, using the machinery of the transference principle and the generalized von Neumann theorem from that paper. A second major component involves nilmanifolds, for which the theory is largely developed from scratch so as to suit the particular needs of the current work.
Overall this is a long and complicated paper, but the exposition is excellent. Given the importance of the results obtained it is well worth detailed study.

11P32 Goldbach-type theorems; other additive questions involving primes
11B30 Arithmetic combinatorics; higher degree uniformity
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