Linear equations in primes

By Ben Joseph Green, Terence Tao. 2006

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Consider a system ψ of nonconstant affine-linear forms ψ 1 , ... , ψ t : 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 ψ 1 (n), ... , ψ t (n) are simultaneously prime. This implies many other well-known 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 affine-linear forms ψ 1 , ... , ψ t 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 Gowers-norm conjecture (GI(s)) and the Mobius and nilsequences conjecture (MN(s)), where s ∈ {1,2,...} is the complexity of the system and measures the extent to which the forms ψ i depend on each other. The case s = 0 is somewhat degenerate, and follows from the prime number theorem in APs. Roughly speaking, the inverse Gowers-norm conjecture GI(s) asserts the Gowers U s+1 -norm of a function f: [N] → [-1, 1] is large if and only if f correlates with an s-step nilsequence, while the Mobius and nilsequences conjecture MN(s) asserts that the Mobius function μ is strongly asymptotically orthogonal to s-step nilsequences of a fixed complexity. These conjectures have long been known to be true for s = 1 (essentially by work of Hardy-Littlewood and Vinogradov), and were established for s = 2 in two papers of the authors. Thus our results in the case of complexity s ≤ 2 are unconditional. In particular we can obtain the expected asymptotics for the number of 4-term progressions p 1 < p 2 < p 3 < p 4 ≤ N of primes, and more generally for any (nondegenerate) problem involving two linear equations in four prime unknowns.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/2887a223909fe5abe60d5f5e32da4805c722a282

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Citations in the corpus

Decompositions, approximate structure, transference, and the Hahn–Banach theorem (2008, positive): “This is a lemma of Green and Tao that we stated as Lemma 4.6.”
Higher order Fourier analysis of multiplicative functions and applications (2014, methodology): “This was established by Landau when deg(P ) = 2 [33] (see also [27]), by Helfgott when deg(P ) = 3 [28, 29], and when P is a product of pairwise independent linear forms by Green, Tao, and Ziegler [22, 23, 24, 25].”
Long gaps between primes (2014, methodology): “The arguments in [12] used recent results [18, 19, 20] on the number of solutions to linear equations in primes, whereas the arguments in [29] instead relied on multidimensional prime-detecting sieves introduced in [27].”
The primes contain arbitrarily long arithmetic progressions (2004, background): “Using this, one may place (a large fraction of) the primes inside a pseudorandom set of “almost primes” (or more precisely, a pseudorandom measure concentrated on almost primes) with positive relative density.”
Almost prime Pythagorean triples in thin orbits (2010, result): “On the other hand, it was recognized in [6] that the recent work of Green and Tao [16] proves the infinitude and Zariski density of the set of all primitive Pythagorean triples x for which FAðxÞ A Pð4Þ, that is, has at most four prime factors.”
An averaged form of Chowla's conjecture (2015, methodology): “, hk over long intervals (of scale comparable to X), one can obtain various averaged forms of the prime tuples conjecture and its relatives, by rather different methods to those used here; see [1], [18], [16], [15], [12].”
Linear Forms and Higher-Degree Uniformity for Functions On $${\mathbb{F}^{n}_{p}}$$ (2010, background): “The decomposition theorem of Green and Tao is inspired by arguments in ergodic theory and proved using averaging projections and energy-increment arguments.”