A quantitative form of the twin prime conjecture asserts that the the number of twin primes less than $n$ is asymptotically equal to $2 C 2\, C\, n/ \ln^2(n)$ where $C$ is the so-called twin prime constant. A variety of sieve methods (originating with Brun) can be used show that the number of twin primes less than $n$ is at most $A A\, n/ \ln^2 (n) $ for some constant $A>2C$. A>2C$. My question is: What is the smallest known value of $A$? I'd also be interested in learning what the best known constants are for the prime k-tuple conjecture?
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What is the best known upper bound for the number of twin primes?A quantitative form of the twin prime conjecture asserts that the the number of twin primes less than $n$ is asymptotically equal to $2 C n/ ln^2(n)$ where $C$ is the so-called twin prime constant. A variety of sieve methods (originating with Brun) can be used show that the number of twin primes less than $n$ is at most $A n/ ln^2 (n) $ for some constant $A>2C$. My question is: What is the smallest known value of $A$? I'd also be interested in learning what the best known constants are for the prime k-tuple conjecture?
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