Let $\mathcal{H}_k = (h_1,h_2,\cdots,h_k)$ be an admissible k-tuple.
The k-tuple conjecture predicts that the number of primes $(b+h_1,b+h_2,\cdots,b+h_k)\in \mathbb{P}^k$ with $b+h_k \leq x$ is:
$$\pi_{\mathcal{H}_k}(x) \sim \left(\prod_{\text{p prime}}\frac{1-\frac{w(\mathcal{H}_k, p)}{p}}{(1-\frac1p)^{k}} \right)\dfrac{x}{\log(x)^k}$$
Where $w(\mathcal{H}_k, p)$ is the number of distinct residues $\pmod p$ in $\mathcal{H}_k$.
We can deduce this formula from Cramer's random model, but Hardy and Littlewood get the same result using the circle method.
How did Hardy and Littlewood formulate this conjecture using the circle method?
