Definition:Twin Primes Constant

Definition

The twin primes constant is the real number:

$\ds \Pi_2$

$:=$

$\ds \prod_{\substack {p \mathop \ge 3 \\ \text {$p$ prime} } } \paren {1 - \dfrac 1 {\paren {p - 1}^2} }$

$\ds $

$\approx$

$\ds 0 \cdotp 66016 \, 18$

$\ds \Pi_2$

$:=$

$\ds 2 \prod_{\substack {p \mathop \ge 3 \\ \text {$p$ prime} } } \paren {1 - \dfrac 1 {\paren {p - 1}^2} }$

$\ds $

$\approx$

$\ds 1 \cdotp 32032 \, 36316 \ldots$

Some sources (in particular François Le Lionnais and Jean Brette, in their Les Nombres Remarquables of $1983$, refer to this as the Shah-Wilson constant.

Research is required to identify who Shah and Wilson were, but the work they reported it in was published around $1919$ in Proceedings of the Cambridge Philosophical Society.

Some sources denote it $C_2$.

Weisstein, Eric W. "Twin Primes Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TwinPrimesConstant.html