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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

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  • $\begingroup$ 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. $\endgroup$
    – Frank Thorne
    Commented Nov 22, 2011 at 20:20
  • $\begingroup$ math.stackexchange.com/questions/4299011/… $\endgroup$
    – Daniel Donnelly
    Commented Nov 7, 2021 at 22:35

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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.

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    $\begingroup$ And, it's worth adding, the fact that this is the only known lower bound isn't because nobody has tried. All modern approaches to proving the twin prime conjecture aim for a quantitative lower bound of the type you describe. $\endgroup$
    – Greg Martin
    Commented Nov 23, 2011 at 0:21
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    $\begingroup$ Greg, I was tempted to quibble this comment. But then it occurred to me that there really aren't any modern approaches to the twin prime conjecture. One could imagine an approach using nonstandard analysis, though, which would give no quantitative lower bounds. $\endgroup$
    – Ben Green
    Commented Nov 26, 2011 at 9:06
  • $\begingroup$ @Ben: yeah, my comment is vague enough to be obviously false (or possibly obviously true, if truly vague). I guess I was trying to communicate to the OP that his idea is part of the "standard" approach to the problem. $\endgroup$
    – Greg Martin
    Commented Dec 8, 2011 at 8:10
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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.

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    $\begingroup$ There is a typo:it should be 2*C_2,not C_2 $\endgroup$
    – Y. Zhao
    Commented Nov 23, 2011 at 4:59

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