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). It is named after William Feller (1906–1970) and Erhard Tornier (1894–1982)
C FT = 1 2 + ( 1 2 ∏ n = 1 ∞ ( 1 − 2 p n 2 ) ) = 1 2 ( 1 + ∏ n = 1 ∞ ( 1 − 2 p n 2 ) ) = 1 2 ( 1 + 1 ζ ( 2 ) ∏ n = 1 ∞ ( 1 − 1 p n 2 − 1 ) ) = 1 2 + 3 π 2 ∏ n = 1 ∞ ( 1 − 1 p n 2 − 1 ) = 0.66131704946 … {\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 Big Omega function is given by
Ω ( x ) = the number of prime factors of x counted by multiplicities {\displaystyle \Omega (x)={\text{the number of prime factors of }}x{\text{ counted by multiplicities}}}
See also: Prime omega function. The Iverson bracket is
[ P ] = { 1 if P is true, 0 if P is false. {\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
C FT = lim n → ∞ ∑ k = 1 n ( [ Ω ( k ) ≡ 0 mod 2 ] ) n {\displaystyle C_{\text{FT}}=\lim _{n\to \infty }{\frac {\sum _{k=1}^{n}([\Omega (k)\equiv 0{\bmod {2}}])}{n}}}
Prime zeta function The prime zeta function P is give by
P ( s ) = ∑ p is prime 1 p s . {\displaystyle P(s)=\sum _{p{\text{ is prime}}}{\frac {1}{p^{s}}}.}
The Feller–Tornier constant satisfies
C FT = 1 2 ( 1 + exp ( − ∑ n = 1 ∞ 2 n P ( 2 n ) n ) ) . {\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