# Is it consistent with Cramer's conjecture to conjecture that $x/\pi_{2}(x)>2B_{2}/6\times\log^{2}x$?

Brun's constant $B_{2}$ is defined as $B_{2}=1/3+1/5+1/5+1/7+1/11+1/13+...$ where the sum is taken on $p$ such that $p$ is an element of a couple of twin primes. The number of twin primes below is expected to be around $2C_{2}x/\log x$ where $C_{2}=0.66016...$ is the twin prime constant. Whenever $(p,p+2)$ is a couple of twin primes, $6\vert (p+1)$ (except if $p=3$).

Now suppose that the maximal gap between consecutive primes of size $x$ is around $\log^{2} x$ (which is consistant with Cramer's conjecture saying $\limsup \dfrac{p_{n+1}-p_{n}}{\log^2 p_{n}}=1$. As numbers $p+1$ such that $(p,p+2)$ is a couple of twin primes represents at most $1/2$ of all primes (that can be of the form $6n+1$ or $6n+5$) and thus $1/6$ of all integers up to $x$, can one expect the number $\pi_{2}(x)$ of twin primes below $x$ to verify $\dfrac{x}{\pi_{2}(x)}>2(1+o(1))\dfrac{B_{2}}{6}\log^{2}x$?

The number of twin primes below $x$ is conjectured to be $\sim 2C_2 x/\log^2 x$.