## Twin Primes Constant

The twin primes constant is defined by

 (1) (2)

where the s in sums and products are taken over Primes only. Flajolet and Vardi (1996) give series with accelerated convergence

 (3) (4)
with

 (5)

where is the Möbius Function. (4) has convergence like .

The most accurately known value of is

 (6)

Le Lionnais (1983, p. 30) calls the Shah-Wilson Constant, and the twin prime constant (Le Lionnais 1983, p. 37).

References

Finch, S. Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/hrdyltl/hrdyltl.html

Flajolet, P. and Vardi, I. Zeta Function Expansions of Classical Constants.'' Unpublished manuscript. 1996. http://pauillac.inria.fr/algo/flajolet/Publications/landau.ps.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983.

Ribenboim, P. The Book of Prime Number Records, 2nd ed. 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.

Wrench, J. W. Evaluation of Artin's Constant and the Twin Prime Constant.'' Math. Comput. 15, 396-398, 1961.