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Statements
H
(
s
)
=
-
ζ
′
(
s
)
+
γ
ζ
(
s
)
+
1
2
ζ
(
s
+
1
)
+
∑
r
=
1
k
ζ
(
1
-
2
r
)
ζ
(
s
+
2
r
)
+
∑
n
=
1
∞
1
n
s
∫
n
∞
B
~
2
k
+
1
(
x
)
x
2
k
+
2
d
x
,
Euler-sum-H
𝑠
diffop
Riemann-zeta
1
𝑠
Riemann-zeta
𝑠
1
2
Riemann-zeta
𝑠
1
superscript
subscript
𝑟
1
𝑘
Riemann-zeta
1
2
𝑟
Riemann-zeta
𝑠
2
𝑟
superscript
subscript
𝑛
1
1
superscript
𝑛
𝑠
superscript
subscript
𝑛
periodic-Bernoulli-polynomial-B
2
𝑘
1
𝑥
superscript
𝑥
2
𝑘
2
𝑥
{\displaystyle{\displaystyle H\left(s\right)=-\zeta'\left(s\right)+\gamma\zeta%
\left(s\right)+\frac{1}{2}\zeta\left(s+1\right)+\sum_{r=1}^{k}\zeta\left(1-2r%
\right)\zeta\left(s+2r\right)+\sum_{n=1}^{\infty}\frac{1}{n^{s}}\int_{n}^{%
\infty}\frac{\widetilde{B}_{2k+1}\left(x\right)}{x^{2k+2}}\mathrm{d}x,}}
γ
{\displaystyle{\displaystyle\gamma}}
H
(
s
)
Euler-sum-H
𝑠
{\displaystyle{\displaystyle H\left(\NVar{s}\right)}}
ζ
(
s
)
Riemann-zeta
𝑠
{\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
∫
{\displaystyle{\displaystyle\int}}
B
~
n
(
x
)
periodic-Bernoulli-polynomial-B
𝑛
𝑥
{\displaystyle{\displaystyle\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)}}
k
𝑘
{\displaystyle{\displaystyle k}}
n
𝑛
{\displaystyle{\displaystyle n}}
x
𝑥
{\displaystyle{\displaystyle x}}
s
𝑠
{\displaystyle{\displaystyle s}}
