# Minimum number of twin primes < N

I saw that someone had calculated an upper bound for the number of twin primes less than a given integer N.

It is obvious that if we can calculate a lower bound for the number of twin primes less than a given integer N, and that lower bound goes to infinity as N goes to infinity, then we have proven that there are infinitely many twin primes.

What research has been done in this direction?

Kermit Rose

• Opera de Cribro, by Friedlander and Iwaniec, is an excellent recent book dealing with this family of problems. You might also see the book by Cojocaru and Murty for a gentler introduction. – Frank Thorne Nov 22 '11 at 20:20

Let $f(N)$ be the number of known twin primes up to $N$. Then I'm afraid the only known lower bound for the number of twin primes up to $N$ is $f(N)$. If no one has ever looked to see whether there are any twin primes between, say, $e^{1000}$ and $e^{2000}$, then for all we know there aren't any.
It is a well-known conjecture that the number of of twin primes below $n$ is asymptotically equal to $2C_2 \frac{n}{(\log n)^2}$ with $C_2 = 0.66...$ (the value of a certain infinite product). See eg this wikipedia page http://en.wikipedia.org/wiki/Twin_prime that also mentions Brun's upper bound which is perhaps what you refer to.