Bounds on the Twin Prime counting function

Question

Maybe I'm mistaken, but wouldn't it suffice to show that the twin prime counting function has any kind of strictly increasing lower bound to show that there are infinitely many twin primes?

Answer 1

I think that you have the answers you want.
But the real answer is no :
$\pi_2{(x)}>\frac{2x-1}{x}$ for every $x>5$
Clearly, $\frac{2x-1}{x}$ is strictly increasing but this tells us nothing for twin primes.

Answer 2

Yes, that would be sufficient.

Answer 3

It would suffice. More precisely, the conjecture is that the twin prime counting function $\pi_2(x)$ has the asymptotics $$ \pi_2(x)\sim 2C_2 \frac{x}{\log (x) ^2}, $$ for $C_2=0.66016...$