란다우 토션트 상수 (Landau's totient constant)[1]
C
L
t
=
∏
p
(
1
+
1
p
(
p
−
1
)
)
=
315
2
π
4
ζ
(
3
)
=
1.943596436820759205057070...
{\displaystyle C_{Lt}=\prod _{p}\left(1+{{1} \over {p(p-1)}}\right)={{315} \over {2\pi ^{4}}}\zeta (3)=1.943596436820759205057070...}
(OEIS 의 수열 A082695 )
C
L
t
=
315
2
π
4
ζ
(
3
)
=
1
ζ
(
2
)
ζ
(
3
)
ζ
(
6
)
=
ζ
(
2
)
ζ
(
3
)
ζ
(
6
)
=
∑
s
=
1
∞
(
μ
(
s
)
)
2
s
ϕ
(
s
)
{\displaystyle C_{Lt}={{315} \over {2\pi ^{4}}}\zeta (3)={{1} \over {{\zeta (2)\zeta (3)} \over {\zeta (6)}}}={{\zeta (2)\zeta (3)} \over {\zeta (6)}}=\sum _{s=1}^{\infty }{{(\mu (s))^{2}} \over {s\phi (s)}}\;\;\;}
[2] [3] [4]
ζ
(
p
)
{\displaystyle \zeta (p)}
리만 제타 함수
,
μ
(
s
)
{\displaystyle ,\;\;\mu (s)}
뫼비우스 함수
,
ϕ
(
s
)
{\displaystyle ,\;\;\phi (s)}
오일러 피 함수
C
L
t
=
{\displaystyle C_{Lt}=}
란다우 토션트 상수
C
L
t
=
∏
p
(
1
+
1
p
(
p
−
1
)
)
{\displaystyle C_{Lt}=\prod _{p}\left(1+{{1} \over {p(p-1)}}\right)}
=
∏
p
(
p
2
−
p
+
1
p
2
−
p
)
{\displaystyle \;\;\;=\prod _{p}\left({{p^{2}-p+1} \over {p^{2}-p}}\right)}
=
∏
p
(
p
2
−
p
p
2
−
p
)
+
(
1
p
2
−
p
)
{\displaystyle \;\;\;=\prod _{p}\left({{p^{2}-p} \over {p^{2}-p}}\right)+\left({{1} \over {p^{2}-p}}\right)}
C
A
=
{\displaystyle C_{A}=}
알틴 상수
C
A
=
∏
p
(
1
−
1
p
(
p
−
1
)
)
{\displaystyle C_{A}=\prod _{p}^{}\left(1-{{1} \over {p(p-1)}}\right)}
=
∏
p
(
p
2
−
p
−
1
p
2
−
p
)
{\displaystyle \;\;\;=\prod _{p}^{}\left({{p^{2}-p-1} \over {p^{2}-p}}\right)}
=
∏
p
(
p
2
−
p
p
2
−
p
)
−
(
1
p
2
−
p
)
{\displaystyle \;\;\;=\prod _{p}\left({{p^{2}-p} \over {p^{2}-p}}\right)-\left({{1} \over {p^{2}-p}}\right)}
C
L
t
=
C
A
+
(
1
p
2
−
p
)
+
(
1
p
2
−
p
)
{\displaystyle C_{Lt}=C_{A}+\left({{1} \over {p^{2}-p}}\right)+\left({{1} \over {p^{2}-p}}\right)}
C
A
=
C
L
t
−
(
1
p
2
−
p
)
−
(
1
p
2
−
p
)
{\displaystyle C_{A}=C_{Lt}-\left({{1} \over {p^{2}-p}}\right)-\left({{1} \over {p^{2}-p}}\right)}
↑ (OEIS A082695), mu(k) is the Möbius function, zeta(z) is the Riemann zeta function, and p_k is the kth prime (Landau 1900; Halberstam and Richert 1974, pp. 110-111; DeKoninck and Ivić 1980, pp. 1-3; Finch 2003, p. 116; Havil 2003, p. 115; Dickson 2005)
↑ Decimal expansion of 315/(2*Pi^4 ) - http://oeis.org/A157292
↑ (Decimal expansion of zeta(6)/(zeta(2)*zeta(3)))http://oeis.org/A068468
↑ (Decimal expansion of zeta(2)*zeta(3)/zeta(6))http://oeis.org/A082695