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Statements
ζ
(
s
)
=
∑
k
=
1
N
1
k
s
+
N
1
-
s
s
-
1
-
1
2
N
-
s
+
∑
k
=
1
n
(
s
+
2
k
-
2
2
k
-
1
)
B
2
k
2
k
N
1
-
s
-
2
k
-
(
s
+
2
n
2
n
+
1
)
∫
N
∞
B
~
2
n
+
1
(
x
)
x
s
+
2
n
+
1
d
x
,
Riemann-zeta
𝑠
superscript
subscript
𝑘
1
𝑁
1
superscript
𝑘
𝑠
superscript
𝑁
1
𝑠
𝑠
1
1
2
superscript
𝑁
𝑠
superscript
subscript
𝑘
1
𝑛
binomial
𝑠
2
𝑘
2
2
𝑘
1
Bernoulli-number-B
2
𝑘
2
𝑘
superscript
𝑁
1
𝑠
2
𝑘
binomial
𝑠
2
𝑛
2
𝑛
1
superscript
subscript
𝑁
periodic-Bernoulli-polynomial-B
2
𝑛
1
𝑥
superscript
𝑥
𝑠
2
𝑛
1
𝑥
{\displaystyle{\displaystyle\zeta\left(s\right)=\sum_{k=1}^{N}\frac{1}{k^{s}}+%
\frac{N^{1-s}}{s-1}-\frac{1}{2}N^{-s}+\sum_{k=1}^{n}\genfrac{(}{)}{0.0pt}{}{s+%
2k-2}{2k-1}\frac{B_{2k}}{2k}N^{1-s-2k}-\genfrac{(}{)}{0.0pt}{}{s+2n}{2n+1}\int%
_{N}^{\infty}\frac{\widetilde{B}_{2n+1}\left(x\right)}{x^{s+2n+1}}\mathrm{d}x,}}
ℜ
s
>
-
2
n
𝑠
2
𝑛
{\displaystyle{\displaystyle\Re s>-2n}}
n
,
N
=
1
,
2
,
3
,
…
formulae-sequence
𝑛
𝑁
1
2
3
…
{\displaystyle{\displaystyle n,N=1,2,3,\dots}}
B
n
Bernoulli-number-B
𝑛
{\displaystyle{\displaystyle B_{\NVar{n}}}}
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)}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
k
𝑘
{\displaystyle{\displaystyle k}}
n
𝑛
{\displaystyle{\displaystyle n}}
x
𝑥
{\displaystyle{\displaystyle x}}
s
𝑠
{\displaystyle{\displaystyle s}}
