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Statements
∑
k
=
1
∞
(
-
1
)
k
k
ζ
(
n
k
,
a
)
=
-
n
ln
Γ
(
a
)
+
ln
(
∏
j
=
0
n
-
1
Γ
(
a
-
e
(
2
j
+
1
)
π
i
/
n
)
)
,
superscript
subscript
𝑘
1
superscript
1
𝑘
𝑘
Hurwitz-zeta
𝑛
𝑘
𝑎
𝑛
Euler-Gamma
𝑎
superscript
subscript
product
𝑗
0
𝑛
1
Euler-Gamma
𝑎
superscript
𝑒
2
𝑗
1
𝜋
𝑖
𝑛
{\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}\zeta\left(nk%
,a\right)=-n\ln\Gamma\left(a\right)+\ln\left(\prod_{j=0}^{n-1}\Gamma\left(a-e^%
{(2j+1)\pi i/n}\right)\right),}}
ℜ
a
≥
1
𝑎
1
{\displaystyle{\displaystyle\Re a\geq 1}}
n
=
2
,
3
,
4
,
…
𝑛
2
3
4
…
{\displaystyle{\displaystyle n=2,3,4,\dots}}
ℜ
a
≥
1
𝑎
1
{\displaystyle{\displaystyle\Re a\geq 1}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
ζ
(
s
,
a
)
Hurwitz-zeta
𝑠
𝑎
{\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}
π
{\displaystyle{\displaystyle\pi}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
ln
z
𝑧
{\displaystyle{\displaystyle\ln\NVar{z}}}
ph
phase
{\displaystyle{\displaystyle\operatorname{ph}}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
k
𝑘
{\displaystyle{\displaystyle k}}
n
𝑛
{\displaystyle{\displaystyle n}}
a
𝑎
{\displaystyle{\displaystyle a}}
Identifiers
