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Title: A maximum likelihood estimation of Brun's constant under the twin primes distribution Authors:  Ryuichi Sawae - Okayama University of Science (Japan) [presenting]
Daisuke Ishii - Okayama University of Science (Japan)
Abstract: The results of research on the distribution of prime numbers are numerous, including a very important theorem in number theory. On the other hand, twin primes are pairs of primes of the form $(p, p+2)$, but it has not been proven that there are infinite number of twins primes until now. Let the set $K_2$ be $\{(3, 5), (5, 7), (11, 13), (17, 19),\ldots\}$ of twin-prime pairs. Then, although the sum of the reciprocals of all the primes is divergent, the sum of the reciprocals of $K_2$, i.e. $B_2 = (1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + (1/17 + 1/19) + \ldots$ , is convergent. The limit of this sum is known as Brun's sum or Brun's constant. For the maximum likelihood estimation of Brun's constant, we list up the twin primes by a fast Eratosthenes sieve, we use a fast computer algorithm for the reciprocals calculation and the error analysis for the Hardy-Littlewood approximation.