Zurück zu
Unendliche Reihen
∑
k
1
,
.
.
.
,
k
6
∈
Z
′
(
−
1
)
k
1
+
.
.
.
+
k
6
(
k
1
2
+
.
.
.
+
k
6
2
)
s
=
4
β
(
s
−
2
)
η
(
s
)
−
16
β
(
s
)
η
(
s
−
2
)
Re
(
s
)
>
1
{\displaystyle {\sum _{k_{1},...,k_{6}\in \mathbb {Z} }}'{\frac {(-1)^{k_{1}+...+k_{6}}}{(k_{1}^{2}+...+k_{6}^{2})^{s}}}=4\,\beta (s-2)\,\eta (s)-16\,\beta (s)\,\eta (s-2)\qquad {\text{Re}}(s)>1}
∑
k
1
,
.
.
.
,
k
8
∈
Z
′
(
−
1
)
k
1
+
.
.
.
+
k
8
(
k
1
2
+
.
.
.
+
k
8
2
)
s
=
−
16
ζ
(
s
)
η
(
s
−
3
)
Re
(
s
)
>
1
{\displaystyle {\sum _{k_{1},...,k_{8}\in \mathbb {Z} }}'{\frac {(-1)^{k_{1}+...+k_{8}}}{(k_{1}^{2}+...+k_{8}^{2})^{s}}}=-16\,\zeta (s)\,\eta (s-3)\qquad {\text{Re}}(s)>1}
This article is issued from
Wikibooks
. The text is licensed under
Creative Commons - Attribution - Sharealike
. Additional terms may apply for the media files.
