Feller–Tornier constant
From HandWiki
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).[1] It is named after William Feller (1906–1970) and Erhard Tornier (1894–1982)[2]
- [math]\displaystyle{ \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]
(sequence A065493 in the OEIS)
Omega function
The Big Omega function is given by
- [math]\displaystyle{ \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]\displaystyle{ [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]\displaystyle{ 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]\displaystyle{ P(s) = \sum_{p \text{ is prime}} \frac 1 {p^s}. }[/math]
The Feller–Tornier constant satisfies
- [math]\displaystyle{ C_\text{FT}= {1\over2} \left( 1+ \exp \left( -\sum_{n=1}^\infty {2^n P(2n) \over n} \right) \right). }[/math]
See also
References
- ↑ "Feller–Tornier Constant – from Wolfram MathWorld". 2017-03-23. http://mathworld.wolfram.com/Feller-TornierConstant.html. Retrieved 2017-03-30.
- ↑ Steven R. Finch. "Mathematical Constants. (Cf. Feller–Tornier constant.)". http://oeis.org/wiki/Feller–Tornier_constant. Retrieved 2017-03-30.
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