Distribution of twin primes in the set of natural numbers (Q1076716)

The author conjectures that \(\pi_2(x)\) is of the order \(\pi(\pi(x))\), where \(\pi(x)\) is the number of primes \(p\leq x\), and \(\pi_2(x)\) is the number of \(p\leq x\) such that \(p+2\) is also a prime. However, it is easily seen that this conjecture is equivalent to \[ C_1x / \log^2x < \pi_2(x) < C_2x / \log^2x \quad (0<C_1<C_2, \quad x\geq x_0). \tag{1} \] The upper bound in (1) is a standard result of sieve theory (this seems to be unknown to the author). The lower bound in (1) (a consequence of the conjectural Hardy-Littlewood asymptotic formula for \(\pi_2(x))\) is one of the most difficult unsolved problems of analytic number theory. The author deduces several consequences of his ``conjecture''. These include (1) (which is obscurely formulated as to appear unconditionally true) and \(\pi_2(x)=o(\pi (x))\) as \(x\to \infty\), which is (unconditionally) trivial.