View source for Feller–Tornier constant

In mathematics, the '''Feller–Tornier constant''' ''C''<sub>FT</sub> 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).<ref>{{cite web|url=http://mathworld.wolfram.com/Feller-TornierConstant.html |title=Feller–Tornier Constant – from Wolfram MathWorld |website=Mathworld.wolfram.com |date=2017-03-23 |accessdate=2017-03-30}}</ref>
It is named after William Feller (1906–1970) and Erhard Tornier (1894–1982)<ref>{{cite web|url=http://oeis.org/wiki/Feller–Tornier_constant|author=Steven R. Finch|title=Mathematical Constants. (Cf. Feller–Tornier constant.)|website=Oeis.org|accessdate=2017-03-30}}</ref>

: <math>
\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}
</math>
{{OEIS|id=A065493}}

==Omega function==

The [[Prime factor#Omega functions|Big Omega function]] is given by

: <math> \Omega(x) = \text{the number of prime factors of } x \text{ counted by multiplicities} </math>

See also: [[Prime omega function]].

The [[Iverson bracket]] is

: <math> [P] = \begin{cases} 1 & \text{if } P \text{ is true,} \\ 0 & \text{if } P \text{ is false.} \end{cases} </math>

With these notations, we have

: <math>C_\text{FT}= \lim_{n\to \infty} \frac{\sum_{k=1}^n ([\Omega(k) \equiv 0 \bmod 2])} {n}</math>

==Prime  zeta  function==

The [[prime zeta function]] ''P'' is give by

: <math> P(s) = \sum_{p \text{ is prime}} \frac 1 {p^s}. </math>

The Feller–Tornier constant satisfies

: <math>C_\text{FT}= {1\over2} \left(  1+ \exp \left( -\sum_{n=1}^\infty {2^n P(2n) \over n} \right) \right).</math>

==See also==
*[[Riemann zeta function]]
*[[L-function]]
*[[Euler product]]
*[[Twin prime]]

==References==
{{Reflist}}

{{DEFAULTSORT:Feller-Tornier constant}}
[[Category:Mathematical constants]]
[[Category:Zeta and L-functions]]
[[Category:Infinite products]]