Twin Primes Constant

The twin primes constant $\Pi_2$ is defined by

$\displaystyle \Pi_2$	$\textstyle \equiv$	$\displaystyle \prod_{\scriptstyle p>2\atop\scriptstyle p{\rm\ prime}} \left[{1-{1\over(p-1)^2}}\right]$	(1)
$\displaystyle \ln({\textstyle{1\over 2}}\Pi_2)$	$\textstyle =$	$\displaystyle \sum_{\scriptstyle p\geq 3\atop\scriptstyle p{\rm\ prime}}\ln\left[{p(p-2)\over (p-1)^2}\right]$
	$\textstyle =$	$\displaystyle \sum_{\scriptstyle p\geq 3\atop\scriptstyle p{\rm\ prime}} \left[{\ln\left({1-{2\over p}}\right)-2\ln\left({1-{1\over p}}\right)}\right]$
	$\textstyle =$	$\displaystyle -\sum_{j=2}^\infty {2^j-2\over j} \sum_{\scriptstyle p\geq 3\atop\scriptstyle p{\rm\ prime}} p^{-j},$	(2)

where the

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

$\Pi_2=\prod_{n=2}^\infty [\zeta(n)(1-2^{-n})]^{-I_n}$ (3)

$={\textstyle{3\over 4}}{\textstyle{15\over 16}}{\textstyle{35\over 36}} \prod_{n=2}^\infty [\zeta(n)(1-2^{-n})(1-3^{-n})(1-5^{-n})(1-7^{-n})]^{-I_n},\quad$ (4)

with

$\begin{displaymath} I_n\equiv {1\over n}\sum_{d\vert n} \mu(d)2^{n/d}, \end{displaymath}$

(5)

where $\mu(x)$ is the Möbius Function. (4) has convergence like $\sim(11/2)^{-n}$ .

The most accurately known value of $\Pi_2$ is

$\begin{displaymath} \Pi_2 = 0.6601618158\ldots. \end{displaymath}$

(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 Unpublished manuscript. 1996.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.

$\Pi_2=\prod_{n=2}^\infty [\zeta(n)(1-2^{-n})]^{-I_n}$	(3)
$={\textstyle{3\over 4}}{\textstyle{15\over 16}}{\textstyle{35\over 36}} \prod_{n=2}^\infty [\zeta(n)(1-2^{-n})(1-3^{-n})(1-5^{-n})(1-7^{-n})]^{-I_n},\quad$	(4)