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).[1] It is named after William Feller (1906–1970) and Erhard Tornier (1894–1982)[2]

${\displaystyle {\begin{aligned}C_{\text{FT}}&={1 \over 2}+\left({1 \over 2}\prod _{n=1}^{\infty }\left(1-{2 \over p_{n}^{2}}\right)\right)\\[4pt]&={{1} \over {2}}\left(1+\prod _{n=1}^{\infty }\left(1-{{2} \over {p_{n}^{2}}}\right)\right)\\[4pt]&={1 \over 2}\left(1+{{1} \over {\zeta (2)}}\prod _{n=1}^{\infty }\left(1-{{1} \over {p_{n}^{2}-1}}\right)\right)\\[4pt]&={1 \over 2}+{{3} \over {\pi ^{2}}}\prod _{n=1}^{\infty }\left(1-{{1} \over {p_{n}^{2}-1}}\right)=0.66131704946\ldots \end{aligned}}}$

(sequence A065493 in the OEIS)

Omega function

The Omega function is given by

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

The Iverson bracket is

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

With these notations, we have

${\displaystyle C_{\text{FT}}=\lim _{n\to \infty }{\frac {\sum _{k=1}^{n}[\Omega (k){\bmod {2}}=0]}{n}}={1 \over 2}}$

Prime zeta function

The prime zeta function P is give by

${\displaystyle P(s)=\sum _{p{\text{ is prime}}}{\frac {1}{p^{s}}}.}$

The Feller–Tornier constant satisfies

${\displaystyle C_{\text{FT}}={1 \over 2}\left(1+\exp \left(-\sum _{n=1}^{\infty }{2^{n}P(2n) \over n}\right)\right).}$

See also

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

References

1. "Feller–Tornier Constant – from Wolfram MathWorld". Mathworld.wolfram.com. 2017-03-23. Retrieved 2017-03-30.
2. Steven R. Finch. "Mathematical Constants. (Cf. Feller–Tornier constant.)". Oeis.org. Retrieved 2017-03-30.