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Statements
(
-
1
)
k
ζ
(
k
)
(
1
-
s
)
=
2
(
2
π
)
s
∑
m
=
0
k
∑
r
=
0
m
(
k
m
)
(
m
r
)
(
ℜ
(
c
k
-
m
)
cos
(
1
2
π
s
)
+
ℑ
(
c
k
-
m
)
sin
(
1
2
π
s
)
)
Γ
(
r
)
(
s
)
ζ
(
m
-
r
)
(
s
)
,
superscript
1
𝑘
Riemann-zeta
𝑘
1
𝑠
2
superscript
2
𝜋
𝑠
superscript
subscript
𝑚
0
𝑘
superscript
subscript
𝑟
0
𝑚
binomial
𝑘
𝑚
binomial
𝑚
𝑟
superscript
𝑐
𝑘
𝑚
1
2
𝜋
𝑠
superscript
𝑐
𝑘
𝑚
1
2
𝜋
𝑠
Euler-Gamma
𝑟
𝑠
Riemann-zeta
𝑚
𝑟
𝑠
{\displaystyle{\displaystyle(-1)^{k}{\zeta^{(k)}}\left(1-s\right)=\frac{2}{(2%
\pi)^{s}}\sum_{m=0}^{k}\sum_{r=0}^{m}\genfrac{(}{)}{0.0pt}{}{k}{m}\genfrac{(}{%
)}{0.0pt}{}{m}{r}\left(\Re(c^{k-m})\cos\left(\tfrac{1}{2}\pi s\right)+\Im(c^{k%
-m})\sin\left(\tfrac{1}{2}\pi s\right)\right){\Gamma^{(r)}}\left(s\right){%
\zeta^{(m-r)}}\left(s\right),}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
ζ
(
s
)
Riemann-zeta
𝑠
{\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
(
m
n
)
binomial
𝑚
𝑛
{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
π
{\displaystyle{\displaystyle\pi}}
cos
z
𝑧
{\displaystyle{\displaystyle\cos\NVar{z}}}
ℑ
absent
{\displaystyle{\displaystyle\Im}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
k
𝑘
{\displaystyle{\displaystyle k}}
m
𝑚
{\displaystyle{\displaystyle m}}
s
𝑠
{\displaystyle{\displaystyle s}}
c
𝑐
{\displaystyle{\displaystyle c}}
Identifiers
