Linear equations in primes

Abstract

Consider a system $\Psi$ of nonconstant affine-linear forms $\psi_1,\dots,\psi_t: \mathbb{Z}^d \to \mathbb{Z}$, no two of which are linearly dependent. Let $N$ be a large integer, and let $K \subseteq [-N,N]^d$ be convex. A generalisation of a famous and difficult open conjecture of Hardy and Littlewood predicts an asymptotic, as $N \to \infty$, for the number of integer points $ n \in \mathbb{Z}^d \cap K$ for which the integers $\psi_1( n),\dots,\psi_t( n)$ 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 $\psi_1,\dots,\psi_t$ 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 (${\rm GI}(s)$) and the Möbius and nilsequences conjecture ($\operatorname{MN}(s)$), where $s \in \{1,2,\dots\}$ 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 $\operatorname{GI}(s)$ asserts the Gowers $U^{s+1}$-norm of a function $f : [N] \rightarrow [-1,1]$ is large if and only if $f$ correlates with an $s$-step nilsequence, while the Möbius and nilsequences conjecture ${\rm MN}(s)$ asserts that the Möbius 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 $s=2$ in two papers of the authors. Thus our results in the case of complexity $s \leq 2$ are unconditional.

In particular we can obtain the expected asymptotics for the number of $4$-term progressions $p_1 < p_2 < p_3 < p_4 \leq N$ of primes, and more generally for any (nondegenerate) problem involving two linear equations in four prime unknowns.