Twin Primes Constant

The twin primes constant  (sometimes also denoted ) is defined by

			(1)
			(2)
			(3)
			(4)

where the s in sums and products are taken over primes only. This can be written as

(5)

where  is the prime zeta function.

Flajolet and Vardi (1996) give series with accelerated convergence

			(6)
			(7)

with

(8)

where  is the Möbius function. The values of  for , 2, ... are 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, ... (OEIS A001037). Equation (7) has convergence like .

 was computed to 45 digits by Wrench (1961) and Gourdon and Sebah list 60 digits.

(9)

(OEIS A005597). Le Lionnais (1983, p. 30) calls  the Shah-Wilson constant, and  the twin prime constant (Le Lionnais 1983, p. 37).

REFERENCES:

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 Unpublished manuscript. 1996. https://algo.inria.fr/flajolet/Publications/landau.ps.

Gourdon, X. and Sebah, P. "Some Constants from Number Theory." https://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.