First Hardy–Littlewood conjecture

Let $P = (m_1, \dots, m_k)$ be a tuple of positive even integers. Let $\pi_P(n)$ denote the number of primes $p\leq n$ such that $(p, p + m_1, \dots, p + m_k)$ forms an admissible prime constellation. Let $w(q; m_1, \dots, m_k)$ denote the number of distinct residues of $0, m_1, \dots, m_k$ modulo $q$, and let

$$C_P = 2 ^ k\prod_{\substack{q\ \text{prime} \\ q\geq 3}} \frac{1 - \frac{w(q; m_1, \dots, m_k)}{q}}{\left(1 - \frac{1}{q}\right)^{k+1}}.$$

Then

$$\pi_P(n)\sim C_P\int_2^n\frac{dt}{\log^{k+1}t}.$$