The generalised Hardy–Littlewood conjecture for affine-linear systems

Let $N$, $d$, $t$, and $L$ be positive integers, and let

$$\Psi:\mathbb{Z}^d\rightarrow\mathbb{Z}^t,\qquad \Psi=(\psi_1,\ldots,\psi_t),$$

be a system of affine-linear forms, none constant and no two rational multiples of one another. Write each form as $\psi_i=\dot{\psi}_i+\psi_i(0)$, let $e_1,\ldots,e_d$ be the standard basis of $\mathbb{Z}^d$, and suppose

$$\|\Psi\|_N:=\sum_{i=1}^t\sum_{j=1}^d|\dot{\psi}_i(e_j)|+\sum_{i=1}^t\left|\frac{\psi_i(0)}{N}\right|\leq L.$$

Let $K\subset[-N,N]^d$ be a convex body, let $P=\{2,3,5,\ldots\}$ be the set of primes, and define

$$\beta_{\infty}=\operatorname{vol}_d\bigl(K\cap\Psi^{-1}(\mathbb{R}_+^t)\bigr).$$

For each prime $p$, let

$$\beta_p=\mathbb{E}_{n\in\mathbb{Z}_p^d}\prod_i\Lambda_{\mathbb{Z}_p}(\psi_i(n)),$$

where the local von Mangoldt function satisfies $\Lambda_{\mathbb{Z}_p}(0)=0$ and $\Lambda_{\mathbb{Z}_p}(b)=p/(p-1)$ for $b\neq0$. Generalised Hardy–Littlewood conjecture. As $N\to\infty$,

$$|K\cap\mathbb{Z}^d\cap\Psi^{-1}(P^t)|=(1+o_{t,d,L}(1))\frac{\beta_{\infty}}{\log^tN}\prod_{p\ \mathrm{prime}}\beta_p+o_{t,d,L}\left(\frac{N^d}{\log^tN}\right).$$

This conjecture generalises the Hardy–Littlewood prediction for prime values of systems of affine-linear forms; the paper applies it to counting prime-entry magic squares and establishes the relevant asymptotics in several dimensions, but the stated general conjecture itself is not resolved here.