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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 $p$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)

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

\Pi_2 = 0.6601618158\ldots.
\end{displaymath} (6)

Le Lionnais (1983, p. 30) calls $C_2$ the Shah-Wilson Constant, and $2C_2$ the twin prime constant (Le Lionnais 1983, p. 37).

See also Brun's Constant, Goldbach Conjecture, Mertens Constant


Finch, S. ``Favorite Mathematical Constants.''

Flajolet, P. and Vardi, I. ``Zeta Function Expansions of Classical Constants.'' Unpublished manuscript. 1996.

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.

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© 1996-9 Eric W. Weisstein