The twin primes constant (sometimes
also denoted ) is defined by
where the s in sums and products are taken over
primes only.
Flajolet and Vardi (1996) give series with accelerated convergence
with
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(8)
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where is the Möbius function. The values of for , 2, ... are
2, 1, 2, 3, 6, 9, 18, 30, 56, 99, ... (Sloane's A001037). (7) has convergence
like .
was computed to 45 digits by Wrench
(1961) and Gourdon and Sebah list 60 digits.
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(9)
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(Sloane's A005597). Le Lionnais (1983, p. 30) calls the Shah-Wilson constant, and the twin prime
constant (Le Lionnais 1983, p. 37).
Finch, S. R. "Hardy-Littlewood Constants." §2.1 in Mathematical Constants. Cambridge, England: Cambridge University
Press, pp. 84-94, 2003.
Flajolet, P. and Vardi, I. "Zeta Function Expansions of Classical Constants." Unpublished manuscript. 1996. http://algo.inria.fr/flajolet/Publications/landau.ps.
Gourdon, X. and Sebah, P. "Some Constants from Number Theory." http://numbers.computation.free.fr/Constants/Miscellaneous/constantsNumTheory.html.
Hardy, G. H. and Littlewood, J. E. "Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes." Acta Math. 44,
1-70, 1923.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983.
Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag,
p. 202, 1989.
Ribenboim, P. The Little Book of Big Primes. New York: Springer-Verlag,
p. 147, 1991.
Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed.
Boston, MA: Birkhäuser, pp. 61-66, 1994.
Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed.
New York: Chelsea, p. 30, 1993.
Sloane, N. J. A. Sequences A001037/M0116 and A005597/M4056 in "The On-Line Encyclopedia of Integer
Sequences."
Wrench, J. W. "Evaluation of Artin's Constant and the Twin Prime Constant."
Math. Comput. 15, 396-398, 1961.
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