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).[1] It is named after William Feller (1906–1970) and Erhard Tornier (1894–1982)[2]
![{\displaystyle {\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)=0.66131704946\ldots \end{aligned}}}](../I/bd188282528754017394645f78a4b0b59dbdf497.svg)
Omega function
The Big Omega function is given by
See also: Prime omega function.
The Iverson bracket is
![{\displaystyle [P]={\begin{cases}1&{\text{if }}P{\text{ is true,}}\\0&{\text{if }}P{\text{ is false.}}\end{cases}}}](../I/54db37a0bfeb6185af816e956c97ee6633a15b62.svg)
With these notations, we have
![{\displaystyle C_{\text{FT}}=\lim _{n\to \infty }{\frac {\sum _{k=1}^{n}([\Omega (k)\equiv 0{\bmod {2}}])}{n}}}](../I/e9c6f6844bc6c018f75fa3c5c3febc84ddc8c44f.svg)
References
- "Feller–Tornier Constant – from Wolfram MathWorld". Mathworld.wolfram.com. 2017-03-23. Retrieved 2017-03-30.
- Steven R. Finch. "Mathematical Constants. (Cf. Feller–Tornier constant.)". Oeis.org. Retrieved 2017-03-30.
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