Feller–Tornier constant

In mathematics, the Feller–Tornier constant C_FT 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)

$\begin{array}{rl}{C}_{\text{FT}}& =\frac{1}{2}+\left(\frac{1}{2}\prod _{n=1}^{\mathrm{\infty }}(1-\frac{2}{p_n^2})\right)\\ & =\frac{1}{2}(1+\prod _{n=1}^{\mathrm{\infty }}(1-\frac{2}{p_n^2}))\\ & =\frac{1}{2}(1+\frac{1}{\zeta (2)}\prod _{n=1}^{\mathrm{\infty }}(1-\frac{1}{p_n^2-1}))\\ & =\frac{1}{2}+\frac{3}{{\pi }^2}\prod _{n=1}^{\mathrm{\infty }}(1-\frac{1}{p_n^2-1})=0.66131704946\dots \end{array}$

(sequence A065493 in the OEIS)

Omega function

The Big Omega function is given by

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

See also: Prime omega function. The Iverson bracket is

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

With these notations, we have

$C_{\text{FT}}=\underset{n\to \mathrm{\infty }}{\lim }\frac{\sum _{k=1}^n([\mathrm{\Omega }(k)\equiv 0mod2])}{n}$

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

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

See also

• Riemann zeta function
• L-function
• Euler product
• Twin prime