Wikipedia, Entziklopedia askea
Matematikan, Feller–Tornier konstantea C FT da 1 baino potentzia handiagoan hazitako faktore primario kopuru bikoitzeko osoko positibo guztien dentsitatea (lehen potentzian soilik agertzen den edozein faktore lehengusu kontuan hartu gabe).[1] William Feller (1906–1970) eta Erhard Tornier (1894–1982) matematikarien izena darama.[2]
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}}}
Omega funtzioa honek ematen du:
Ω
(
x
)
=
x
-en faktore lehenenn kopurua multiplizitateen arabera kontatuta
{\displaystyle \Omega (x)=x{\text{-en faktore lehenenn kopurua multiplizitateen arabera kontatuta}}}
Iversonen parentesia
[
P
]
=
{
1
baldin
P
egia,
0
baldin
P
gezurra.
{\displaystyle [P]={\begin{cases}1&{\text{baldin }}P{\text{ egia,}}\\0&{\text{baldin }}P{\text{ gezurra.}}\end{cases}}}
Notazio hauekin,
C
FT
=
lim
n
→
∞
∑
k
=
1
n
[
Ω
(
k
)
mod
2
=
0
]
n
=
1
2
{\displaystyle C_{\text{FT}}=\lim _{n\to \infty }{\frac {\sum _{k=1}^{n}[\Omega (k){\bmod {2}}=0]}{n}}={1 \over 2}}
Zeta lehenaren funtzioa honek ematen du:
P
(
s
)
=
∑
p
lehena bada
1
p
s
.
{\displaystyle P(s)=\sum _{p{\text{ lehena bada}}}{\frac {1}{p^{s}}}.}
Feller-Tornier konstanteak hau betetzen du:
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).}
![{\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/654f4bee7410a10fc164b081ef22a6e86c56b69c)
![{\displaystyle [P]={\begin{cases}1&{\text{baldin }}P{\text{ egia,}}\\0&{\text{baldin }}P{\text{ gezurra.}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/084122d78ad3c0d2532aafbadf6a094d27740e70)
![{\displaystyle C_{\text{FT}}=\lim _{n\to \infty }{\frac {\sum _{k=1}^{n}[\Omega (k){\bmod {2}}=0]}{n}}={1 \over 2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a59647ed79728ba0dc495fb54e81bcdecc6a337)
Datuak: Q29048688