Feller–Tornier constant

In mathematics, the Feller–Tornier constant CFT is the density of the set of all positive integers that have an even number of distinct prime factors raised to a power larger than one (ignoring any prime factors which appear only to the first power). It is named after William Feller (1906–1970) and Erhard Tornier (1894–1982)



\begin{align} C_\text{FT} & ={1\over2}+\left( {1\over2} \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\over2}\left(1+{{1}\over{\zeta(2)}} \prod_{n=1}^\infty \left( 1-{{1}\over{p_n^2 -1}} \right) \right) \\[4pt] & = {1\over2}+{{3}\over{\pi^2}} \prod_{n=1}^\infty \left( 1-{{1} \over {p_n^2 -1}} \right)= 0.66131704946\ldots \end{align} $$

Omega function
The Big Omega function is given by


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

See also: Prime omega function.

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


 * $$C_\text{FT}= \lim_{n\to \infty} \frac{\sum_{k=1}^n ([\Omega(k) \equiv 0 \bmod 2])} {n}$$

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


 * $$C_\text{FT}= {1\over2} \left( 1+ \exp \left( -\sum_{n=1}^\infty {2^n P(2n) \over n} \right) \right).$$