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

			(1)
			(2)
		{sum_(p>2; p prime)ln[(p(p-2))/((p-1)^2)]}" src="https://mathworld.wolfram.com/images/equations/TwinPrimesConstant/Inline11.gif" style="height:79px; width:152px" />	(3)
		{sum_(p>2; p prime)[ln(1-2/p)-2ln(1-1/p)]}," src="https://mathworld.wolfram.com/images/equations/TwinPrimesConstant/Inline14.gif" style="height:79px; width:234px" />	(4)

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

{(2-2^n)/n[P(n)-2^(-n)]}, " src="https://mathworld.wolfram.com/images/equations/TwinPrimesConstant/NumberedEquation1.gif" style="height:36px; width:190px" />

(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 Unpublished manuscript. 1996.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.