Hardy–Littlewood twin-prime asymptotic conjecture ($\#\{p\le N:p+2\text{ prime}\}\sim 2C_{2}N/(\log N{)}^2$)

The local-density prediction for twin primes is the displayed asymptotic, with the twin-prime constant $C_2$. It is stronger than infinitude alone. The Selberg sieve proves the correct order of magnitude as an upper bound, but the parity problem in sieve theory prevents its elementary majorant from proving this conjectured asymptotic.

// Target: analytic-number-theory.bigb