K
=
∏
k
=
1
∞
(
1
+
1
k
2
+
2
k
)
log
2
k
=
∏
k
=
1
∞
k
log
2
(
1
+
1
k
2
+
2
k
)
{\displaystyle K=\prod _{k=1}^{\infty }{\left(1+{1 \over k^{2}+2k}\right)}^{\log _{2}k}=\prod _{k=1}^{\infty }k^{\log _{2}^{}\left(1+{1 \over k^{2}+2k}\right)}}
K
=
∏
k
=
1
∞
(
1
+
1
k
(
k
+
2
)
)
log
2
k
=
∏
k
=
1
∞
(
1
+
1
k
(
k
+
2
)
)
l
n
k
l
n
2
{\displaystyle K=\prod _{k=1}^{\infty }{\left(1+{1 \over k(k+2)}\right)}^{\log _{2}k}=\prod _{k=1}^{\infty }{\left(1+{1 \over k(k+2)}\right)}^{{lnk} \over {ln2}}}
K
=
1
log
(
2
)
∑
s
=
1
∞
ζ
(
2
s
)
−
1
s
∑
k
=
1
2
s
−
1
−
1
(
k
+
1
)
k
{\displaystyle K={{1} \over {\log(2)}}\sum _{s=1}^{\infty }{{\zeta (2s)-1} \over {s}}\sum _{k=1}^{2s-1}{{-1^{(k+1)}} \over {k}}}
K
=
e
x
p
(
1
l
n
2
∑
k
=
1
∞
H
2
k
−
1
′
(
ζ
(
2
k
)
−
1
)
k
)
{\displaystyle K=exp\left({{1} \over {ln2}}\sum _{k=1}^{\infty }{{H_{2k-1}^{'}(\zeta (2k)-1)} \over {k}}\right)}
H
{\displaystyle H}
는 조화수 ,
ζ
{\displaystyle \zeta }
리만 제타 함수
log
(
K
0
)
=
1
log
(
2
)
∑
s
=
1
∞
ζ
(
2
s
)
−
1
s
∑
k
=
1
2
s
−
1
−
1
(
k
+
1
)
k
{\displaystyle \log(K_{0})={{1} \over {\log(2)}}\sum _{s=1}^{\infty }{{\zeta (2s)-1} \over {s}}\sum _{k=1}^{2s-1}{{-1^{(k+1)}} \over {k}}}
log
(
K
0
)
log
(
2
)
=
∑
s
=
1
∞
ζ
(
2
s
)
−
1
s
∑
k
=
1
2
s
−
1
−
1
(
k
+
1
)
k
{\displaystyle \log(K_{0}){\log(2)}=\sum _{s=1}^{\infty }{{\zeta (2s)-1} \over {s}}\sum _{k=1}^{2s-1}{{-1^{(k+1)}} \over {k}}}
log
(
K
0
)
log
(
2
)
=
∑
s
=
1
∞
ζ
(
2
s
)
−
1
s
(
1
1
−
1
2
+
1
3
−
⋯
+
1
(
2
s
−
1
)
)
{\displaystyle \log(K_{0}){\log(2)}=\sum _{s=1}^{\infty }{{\zeta (2s)-1} \over {s}}\left({1 \over 1}-{1 \over 2}+{1 \over 3}-\cdots +{1 \over (2s-1)}\right)}