I got that idea.........in part from Euler's product......

Notable constants[edit]

Many well known constants have Euler product expansions.
The Leibniz formula for π,

${\displaystyle \pi /4=\sum _{n=0}^{\infty }\,{\frac {(-1)^{n}}{2n+1}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots ,}$

can be interpreted as a Dirichlet series using the (unique) Dirichlet character modulo 4, and converted to an Euler product of superparticular ratios

${\displaystyle \pi /4=\left(\prod _{p\equiv 1{\pmod {4}}}{\frac {p}{p-1}}\right)\cdot \left(\prod _{p\equiv 3{\pmod {4}}}{\frac {p}{p+1}}\right)={\frac {3}{4}}\cdot {\frac {5}{4}}\cdot {\frac {7}{8}}\cdot {\frac {11}{12}}\cdot {\frac {13}{12}}\cdots ,}$

where each numerator is a prime number and each denominator is the nearest multiple of four.^[1]
Other Euler products for known constants include:
Hardy–Littlewood's twin prime constant:

${\displaystyle \prod _{p>2}{\Big (}1-{\frac {1}{(p-1)^{2}}}{\Big )}=0.660161...}$