Twin prime

Two primes differing by two form a twin-prime pair, for example $3,5$ or $11,13$. The twin-prime upper bound using Bombieri–Vinogradov limits their number without proving there are infinitely many.

The conjecture asserts that infinitely many prime numbers $p$ have $p+2$ prime. The Hardy–Littlewood twin-prime asymptotic conjecture predicts their quantitative frequency and is stronger. The Selberg sieve supplies an upper bound of the predicted order but does not prove infinitude.

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

This positive convergent product corrects for the dependence of primality of $n$ and $n+2$ at each odd prime. The associated singular series is $2C_2$: the factor at two is two, and at an odd prime the local correction is $(1-2/p)/(1-1/p{)}^2$. It appears in the Hardy–Littlewood twin-prime asymptotic conjecture and in upper bounds from the Selberg sieve.

// Target: analytic-number-theory.bigb

The sum of reciprocals of twin primes converges, whether or not there are infinitely many twin primes. The twin-prime upper bound from a quadratic sieve and reciprocal-sum convergence from a counting bound prove it. This contrasts with the Euler proof that the sum of reciprocals of primes diverges.

Apply the Selberg upper-bound sieve to the numbers $p+2$ with $p\le x$ prime. For odd squarefree integers $d$, the local count is $\pi (x;d,-2)$ and the density is $1/\varphi (d)$. With sieve level $D=x^{1/3}$, the normalizing sum is $\gg \log x$ by the truncated Euler-product lower bound. The Selberg least-common-multiple weights have absolute values at most $3^{\omega (d)}$. The weighted arithmetic-progression error bound and Bombieri–Vinogradov theorem make their total error $O(x/{\log }^{3}x)$. Consequently the number of twin prime pairs with smaller member at most $x$ is $O(x/{\log }^{2}x)$.