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Statements
H
(
s
)
=
1
2
ζ
(
s
+
1
)
+
ζ
(
s
)
s
-
1
-
∑
r
=
1
k
(
s
+
2
r
-
2
2
r
-
1
)
ζ
(
1
-
2
r
)
ζ
(
s
+
2
r
)
-
(
s
+
2
k
2
k
+
1
)
∑
n
=
1
∞
1
n
∫
n
∞
B
~
2
k
+
1
(
x
)
x
s
+
2
k
+
1
d
x
.
Euler-sum-H
𝑠
1
2
Riemann-zeta
𝑠
1
Riemann-zeta
𝑠
𝑠
1
superscript
subscript
𝑟
1
𝑘
binomial
𝑠
2
𝑟
2
2
𝑟
1
Riemann-zeta
1
2
𝑟
Riemann-zeta
𝑠
2
𝑟
binomial
𝑠
2
𝑘
2
𝑘
1
superscript
subscript
𝑛
1
1
𝑛
superscript
subscript
𝑛
periodic-Bernoulli-polynomial-B
2
𝑘
1
𝑥
superscript
𝑥
𝑠
2
𝑘
1
𝑥
{\displaystyle{\displaystyle H\left(s\right)=\frac{1}{2}\zeta\left(s+1\right)+%
\frac{\zeta\left(s\right)}{s-1}-\sum_{r=1}^{k}\genfrac{(}{)}{0.0pt}{}{s+2r-2}{%
2r-1}\zeta\left(1-2r\right)\zeta\left(s+2r\right)-\genfrac{(}{)}{0.0pt}{}{s+2k%
}{2k+1}\sum_{n=1}^{\infty}\frac{1}{n}\int_{n}^{\infty}\frac{\widetilde{B}_{2k+%
1}\left(x\right)}{x^{s+2k+1}}\mathrm{d}x.}}
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)}}
(
m
n
)
binomial
𝑚
𝑛
{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
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}}
