Linear equations in primes

• 成果类型: Article
• 署名作者: Green, Benjamin; Tao, Terence
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• 发表日期: 2010
• 页码: 1753-1850
• 关键词: arithmetic progressions; mobius function; THEOREM; UNIFORMITY; number

摘要

Consider a system Psi of nonconstant affine-linear forms psi(1),...,psi(t) : Z(d) -> Z, no two of which are linearly dependent. Let N be a large integer, and let K subset of [-N, N](d) be convex. A generalisation of a famous and difficult open conjecture of Hardy and Littlewood predicts an asymptotic, as N -> infinity, for the number of integer points n is an element of Z(d) boolean AND K for which the integers psi(1)(n),...,psi(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 psi(1),...,psi(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 is an element of {1, 2,...} is the complexity of the system and measures the extent to which the forms psi(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 Us+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 mu 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 D 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.