# Feller–Tornier constant

In mathematics, the Feller–Tornier constant CFT is the density of the set of all integers that have an even number of prime factors (counted by multiplicities). It is named after William Feller (1906–1970) and Erhard Tornier (1894–1982)

{\begin{aligned}C_{\text{FT}}&={1 \over 2}+\left({1 \over 2}\prod _{n=1}^{\infty }\left(1-{2 \over p_{n}^{2}}\right)\right)\\[4pt]&={{1} \over {2}}\left(1+\prod _{n=1}^{\infty }\left(1-{{2} \over {p_{n}^{2}}}\right)\right)\\[4pt]&={1 \over 2}\left(1+{{1} \over {\zeta (2)}}\prod _{n=1}^{\infty }\left(1-{{1} \over {p_{n}^{2}-1}}\right)\right)\\[4pt]&={1 \over 2}+{{3} \over {\pi ^{2}}}\prod _{n=1}^{\infty }\left(1-{{1} \over {p_{n}^{2}-1}}\right)=0.661317\ldots \end{aligned}} (sequence A065493 in the OEIS)

## Omega function

The Omega function is given by

$\Omega (x)={\text{the number of prime factors of }}x{\text{ counted by multiplicities}}$

The Iverson bracket is

$[P]={\begin{cases}1&{\text{if }}P{\text{ is true,}}\\0&{\text{if }}P{\text{ is false.}}\end{cases}}$

With these notations, we have

$C_{\text{FT}}=\lim _{n\to \infty }{\frac {\sum _{k=1}^{n}[\Omega (k){\bmod {2}}=0]}{n}}={1 \over 2}$

## Prime zeta function

The prime zeta function P is give by

$P(s)=\sum _{p{\text{ is prime}}}{\frac {1}{p^{s}}}.$

The Feller–Tornier constant satisfies

$C_{\text{FT}}={1 \over 2}\left(1+\exp \left(-\sum _{n=1}^{\infty }{2^{n}P(2n) \over n}\right)\right).$