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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd188282528754017394645f78a4b0b59dbdf497)
(sequence A065493 in the OEIS)
Omega function
The Big Omega function is given by
![{\displaystyle \Omega (x)={\text{the number of prime factors of }}x{\text{ counted by multiplicities}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3da9a4dc90e9c0c4c8c7352a5ba85d8a5bf342c8)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54db37a0bfeb6185af816e956c97ee6633a15b62)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9c6f6844bc6c018f75fa3c5c3febc84ddc8c44f)
Prime zeta function
The prime zeta function P is give by
![{\displaystyle P(s)=\sum _{p{\text{ is prime}}}{\frac {1}{p^{s}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1f4ad9081e33865b6f1e87ca47b39056bd280c1)
The Feller–Tornier constant satisfies
![{\displaystyle C_{\text{FT}}={1 \over 2}\left(1+\exp \left(-\sum _{n=1}^{\infty }{2^{n}P(2n) \over n}\right)\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f4739b1271d8bd6b785e1a60ba2940b5ed0ab3a)
See also
- Riemann zeta function
- L-function
- Euler product
- Twin prime
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.