The sums-of-two-squares $k$-tuple conjecture

Let $\mathbb{S}$ denote the set of integers expressible as a sum of two squares. For $k\geq 1$ and a set $\boldsymbol{h}=\{h_1,\ldots,h_k\}\subseteq\mathbb{Z}$ with $\lvert\boldsymbol{h}\rvert=k$, let

$$R_k(\boldsymbol{h};x)=\frac{1}{x}\sum_{n\leq x}\mathbf{1}_{\mathbb{S}}(n+h_1)\cdots\mathbf{1}_{\mathbb{S}}(n+h_k),$$

where $R_1(x)=\lvert\mathbb{S}\cap[1,x]\rvert/x$, and let $\mathfrak{S}_{\boldsymbol{h}}$ be the singular series defined from the local densities $\delta_{\boldsymbol{h}}(p)$. The sums-of-two-squares $k$-tuple conjecture. For fixed $k\geq 1$ and $\boldsymbol{h}=\{h_1,\ldots,h_k\}\subseteq\mathbb{Z}$ with $\lvert\boldsymbol{h}\rvert=k$, if $\mathfrak{S}_{\boldsymbol{h}}>0$, then

$$R_k(\boldsymbol{h};x)\sim \mathfrak{S}_{\boldsymbol{h}}\bigl(R_1(x)\bigr)^k\qquad(x\to\infty).$$

This is the analogue for sums of two squares of the Hardy–Littlewood prime $k$-tuple conjecture; the paper presents evidence but does not establish the asymptotic in general.