黎曼泽塔函数 ，写作ζ(s ) 复数 s Re(s ) > 1
ζ
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s
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=
∑
n
=
1
∞
1
n
s
{\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}}
复平面 中一矩形区域之黎曼ζ函数
ζ
(
z
)
{\displaystyle \zeta (z)}
Matplotlib 程式绘图产生，使用到定义域着色 方法。[ 1] 它亦可以用积分定义：
ζ
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s
)
=
1
Γ
(
s
)
∫
0
∞
x
s
−
1
e
x
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1
d
x
{\displaystyle \zeta (s)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {x^{s-1}}{e^{x}-1}}\mathrm {d} x}
在区域 {s : Re(s ) > 1} 无穷级数 收敛并为一全纯函数 。欧拉 在1740年考虑过 s 切比雪夫 拓展到 s > 1[ 2] 波恩哈德·黎曼 认识到：ζ函数可以通过解析延拓 ，把定义域 扩展到几乎整个复数域上的全纯函数 ζ(s ) 黎曼猜想 所研究的函数。
虽然黎曼的ζ函数被数学家认为主要和“最纯”的数学领域数论 相关，它也出现在应用统计学 （参看齐夫定律 和齐夫-曼德尔布罗特定律 物理 ，以及调音 的数学理论中。
ζ函数最早出现于1350年左右，尼克尔·奥里斯姆 发现了调和级数 发散，即：
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→
∞
{\displaystyle \zeta (1)=1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+...\to \infty }
More information
奥里斯姆对调和级数发散的“证明”
ζ
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{\displaystyle {\begin{aligned}\zeta (1)&=1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{8}}+...\\&=1+{\frac {1}{2}}+({\frac {1}{3}}+{\frac {1}{4}})+({\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{8}})+...\\&\geq 1+{\frac {1}{2}}+({\frac {1}{4}}+{\frac {1}{4}})+({\frac {1}{8}}+{\frac {1}{8}}+{\frac {1}{8}}+{\frac {1}{8}})+...\\&=1+{\frac {1}{2}}+{\frac {1}{2}}+{\frac {1}{2}}+...\\&\to \infty \\\end{aligned}}}
Close
第n个调和数（蓝点）与Log(n)+γ(红线)的图像 之后的一次进展来自莱昂哈德·欧拉 ，他给出了调和级数呈对数发散。
More information
欧拉对调和级数发散速度的证明[ 3]
为了求出调和级数的部分和，使用欧拉-麦克劳林求和公式 （当然，亦可使用阿贝尔求和公式 ）：
∑
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{\displaystyle \sum _{y<n\leq x}f(n)=\int _{y}^{x}f(t)\,\mathrm {d} t+\int _{y}^{x}(t-\left\lfloor t\right\rfloor )f'(t)\,\mathrm {d} t+f(x)(\left\lfloor x\right\rfloor -x)-f(y)(\left\lfloor y\right\rfloor -y)}
∑
n
≤
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1
n
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+
∫
1
x
1
t
d
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∫
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⌊
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⌋
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t
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x
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x
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ln
x
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1
x
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t
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⌊
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{\displaystyle {\begin{aligned}\sum _{n\leq x}{\frac {1}{n}}&=1+\int _{1}^{x}{\frac {1}{t}}\,\mathrm {d} t-\int _{1}^{x}{\frac {(t-\left\lfloor t\right\rfloor )}{t^{2}}}\,\mathrm {d} t+{\frac {\left\lfloor x\right\rfloor -x}{x}}\\&=1+\ln x-\int _{1}^{x}{\frac {(t-\left\lfloor t\right\rfloor )}{t^{2}}}\,\mathrm {d} t+\mathrm {O} \left({\frac {1}{x}}\right)\\&=1+\ln x-\int _{1}^{\infty }{\frac {(t-\left\lfloor t\right\rfloor )}{t^{2}}}\,\mathrm {d} t+\int _{x}^{\infty }{\frac {(t-\left\lfloor t\right\rfloor )}{t^{2}}}\,\mathrm {d} t+\mathrm {O} \left({\frac {1}{x}}\right)\\&=\ln x+1-\int _{1}^{\infty }{\frac {(t-\left\lfloor t\right\rfloor )}{t^{2}}}\,\mathrm {d} t+\int _{x}^{\infty }{\frac {\left\{t\right\}}{t^{2}}}\,\mathrm {d} t+\mathrm {O} \left({\frac {1}{x}}\right)\\\end{aligned}}}
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t
{\displaystyle 1-\int _{1}^{\infty }{\frac {(t-\left\lfloor t\right\rfloor )}{t^{2}}}\,\mathrm {d} t}
欧拉-马斯刻若尼常数 γ
再考虑剩下的一个积分，也就是
∫
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{
t
}
t
2
d
t
{\displaystyle \int _{x}^{\infty }{\frac {\left\{t\right\}}{t^{2}}}\,\mathrm {d} t}
{
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{\displaystyle \left\{t\right\}\leq 1}
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{\displaystyle \int _{x}^{\infty }{\frac {\left\{t\right\}}{t^{2}}}\,\mathrm {d} t\leq \int _{x}^{\infty }{\frac {1}{t^{2}}}\,\mathrm {d} t={\frac {1}{x}}}
∑
n
≤
x
1
n
=
ln
x
+
γ
+
O
(
1
x
)
{\displaystyle \sum _{n\leq x}{\frac {1}{n}}=\ln x+\gamma +\mathrm {O} ({\frac {1}{x}})}
Close
除此之外，他还在1735年给出了巴塞尔问题 的解答，得到
ζ
(
2
)
=
π
2
6
{\displaystyle \zeta (2)={\frac {\pi ^{2}}{6}}}
巴塞尔问题#欧拉的错误证明 中看到，然而那是他的第一个证明，因而广为人知。[ 4]
More information 欧拉对
欧拉对
ζ
(
2
)
=
π
2
6
{\displaystyle {\begin{smallmatrix}\zeta (2)={\frac {\pi ^{2}}{6}}\end{smallmatrix}}}
下面将写出欧拉对上式的证明中缺失的严格论证的部分，即对连乘积公式的证明部分，而不涉及最终的系数比较
首先考虑当n为奇数时，将
z
n
−
a
n
{\displaystyle z^{n}-a^{n}}
a
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a
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n
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n
{\displaystyle a,ae^{2\pi i{\frac {1}{n}}},ae^{2\pi i{\frac {2}{n}}},...,ae^{2\pi i{\frac {n-1}{n}}}}
由于n为奇数，所以可以将除了z=a外的其他根及其共轭一一配对，即 将共轭的根一一配对
a
e
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k
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e
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k
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a
e
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k
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{\displaystyle ae^{2\pi i{\frac {k}{n}}},ae^{2\pi i{\frac {n-k}{n}}}=ae^{-2\pi i{\frac {k}{n}}}}
韦达定理 可以还原出每对根的最小多项式：
x
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x
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a
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a
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a
e
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k
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cos
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cos
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−
2
π
k
n
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=
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cos
(
2
π
k
n
)
{\displaystyle x_{1}+x_{2}=-{\frac {a_{1}}{a_{0}}}=ae^{2\pi i{\frac {k}{n}}}+ae^{-2\pi i{\frac {k}{n}}}=\cos \left(2\pi {\frac {k}{n}}\right)+\cos \left(-2\pi {\frac {k}{n}}\right)=2\cos \left({\frac {2\pi k}{n}}\right)}
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1
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a
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k
n
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{\displaystyle x_{1}x_{2}={\frac {a_{2}}{a_{0}}}=ae^{2\pi i{\frac {k}{n}}}ae^{-2\pi i{\frac {k}{n}}}=a^{2}}
a
0
=
1
{\displaystyle a_{0}=1}
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+
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2
−
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cos
(
2
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k
n
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+
a
2
{\displaystyle a_{0}z^{2}+a_{1}z+a_{2}=z^{2}-2\cos \left({\tfrac {2\pi k}{n}}\right)z+a^{2}}
n
−
1
2
{\displaystyle {\tfrac {n-1}{2}}}
z
−
a
{\displaystyle z-a}
z
n
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(
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{\displaystyle z^{n}-a^{n}=(z-a)\prod _{k=1}^{\frac {n-1}{2}}\left(z^{2}-2az\cos {\frac {2k\pi }{n}}+a^{2}\right)}
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{\displaystyle z=1+{\frac {x}{N}},a=1-{\frac {x}{N}},N=n}
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cos
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+
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cos
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=
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N
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(
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cos
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{
[
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[
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cos
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}
{\displaystyle {\begin{aligned}\left(1+{\frac {x}{N}}\right)^{N}-\left(1-{\frac {x}{N}}\right)^{N}&=\left[\left(1+{\frac {x}{N}}\right)-\left(1-{\frac {x}{N}}\right)\right]\prod _{k=1}^{\frac {N-1}{2}}\left[\left(1+{\frac {x}{N}}\right)^{2}-2\left(1+{\frac {x}{N}}\right)\left(1-{\frac {x}{N}}\right)\cos \left({\frac {2\pi k}{N}}\right)+\left(1-{\frac {x}{N}}\right)^{2}\right]\\&={\frac {2x}{N}}\prod _{k=1}^{\frac {N-1}{2}}\left[2+{\frac {2x^{2}}{N^{2}}}-2\left(1-{\frac {x^{2}}{N^{2}}}\right)\cos \left({\frac {2\pi k}{N}}\right)\right]\\&={\frac {2x}{N}}\prod _{k=1}^{\frac {N-1}{2}}\left[{2+{\frac {2{x^{2}}}{N^{2}}}-2\cos \left({\frac {2\pi k}{N}}\right)+{\frac {2{x^{2}}}{N^{2}}}\cos({\frac {2\pi k}{N}})}\right]\\&={\frac {4x}{N}}\prod _{k=1}^{\frac {N-1}{2}}\left({(1-\cos({\frac {2\pi k}{N}}))+(1+\cos({\frac {2\pi k}{N}})){\frac {x^{2}}{N^{2}}}}\right)\\&={\frac {4x}{N}}\prod _{k=1}^{\frac {N-1}{2}}\left\{\left[1-\cos \left({\frac {2\pi k}{N}}\right)\right]\left[{1+{\frac {1+\cos \left({\frac {2\pi k}{N}}\right)}{1-\cos({\frac {2\pi k}{N}})}}{\frac {x^{2}}{N^{2}}}}\right]\right\}\\\end{aligned}}}
4
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{\displaystyle {\frac {4}{N}}\prod _{k=1}^{\frac {N-1}{2}}(1-\cos({\frac {2\pi k}{N}}))}
C
(
N
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{\displaystyle C(N)}
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x
2
N
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{\displaystyle \left(1+{\frac {x}{N}}\right)^{N}-\left(1-{\frac {x}{N}}\right)^{N}={C(N)}x\prod _{k=1}^{\frac {N-1}{2}}\left({1+{\frac {1+\cos({\frac {2\pi k}{N}})}{1-\cos({\frac {2\pi k}{N}})}}{\frac {x^{2}}{N^{2}}}}\right)}
(
1
+
x
N
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N
=
∑
k
=
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N
C
N
k
x
k
N
k
{\displaystyle {(1+{\frac {x}{N}})^{N}}=\sum _{k=0}^{N}{C_{N}^{k}{\frac {x^{k}}{N^{k}}}}}
(
1
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N
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N
=
∑
k
=
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N
(
−
1
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k
C
N
k
x
k
N
k
{\displaystyle {(1-{\frac {x}{N}})^{N}}=\sum _{k=0}^{N}{{{(-1)}^{k}}C_{N}^{k}{\frac {x^{k}}{N^{k}}}}}
C
N
1
x
N
−
(
−
1
)
C
N
1
x
N
=
2
C
N
1
x
N
=
2
x
{\displaystyle C_{N}^{1}{\frac {x}{N}}-(-1)C_{N}^{1}{\frac {x}{N}}=2C_{N}^{1}{\frac {x}{N}}=2x}
C
(
N
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=
2
{\displaystyle C(N)=2}
(
1
+
x
N
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N
−
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1
−
x
N
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N
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2
x
∏
k
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2
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1
+
1
+
cos
(
2
π
k
N
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1
−
cos
(
2
π
k
N
)
x
2
N
2
)
{\displaystyle \left(1+{\frac {x}{N}}\right)^{N}-\left(1-{\frac {x}{N}}\right)^{N}=2x\prod _{k=1}^{\frac {N-1}{2}}\left({1+{\frac {1+\cos({\frac {2\pi k}{N}})}{1-\cos({\frac {2\pi k}{N}})}}{\frac {x^{2}}{N^{2}}}}\right)}
θ
=
2
π
k
N
{\displaystyle \theta ={\frac {2\pi k}{N}}}
cos
(
θ
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=
1
−
θ
2
2
+
O
(
θ
3
)
{\displaystyle \cos(\theta )=1-{\frac {\theta ^{2}}{2}}+\mathrm {O} (\theta ^{3})}
(
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N
)
N
−
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1
−
x
N
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N
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2
x
∏
k
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−
1
2
[
1
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1
+
cos
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2
π
k
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1
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cos
(
2
π
k
N
)
x
2
N
2
]
=
2
x
∏
k
=
1
N
−
1
2
{
1
+
1
+
[
1
−
θ
2
2
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O
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θ
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)
]
1
−
[
1
−
θ
2
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+
O
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θ
3
)
]
x
2
N
2
}
=
2
x
∏
k
=
1
N
−
1
2
[
1
+
2
−
θ
2
2
+
O
(
θ
3
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θ
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+
O
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θ
3
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x
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N
2
]
=
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x
∏
k
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1
N
−
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2
(
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(
4
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O
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θ
3
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)
x
2
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θ
2
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O
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θ
3
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N
2
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=
2
x
∏
k
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1
N
−
1
2
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1
+
(
4
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k
π
N
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2
+
O
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x
2
(
(
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k
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2
+
O
(
(
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3
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)
N
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2
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2
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(
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2
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π
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)
)
{\displaystyle {\begin{aligned}\left(1+{\frac {x}{N}}\right)^{N}-\left(1-{\frac {x}{N}}\right)^{N}&=2x\prod _{k=1}^{\frac {N-1}{2}}\left[1+{\frac {1+\cos \left({\frac {2\pi k}{N}}\right)}{1-\cos \left({\frac {2\pi k}{N}}\right)}}{\frac {x^{2}}{N^{2}}}\right]\\&=2x\prod _{k=1}^{\frac {N-1}{2}}\left\{1+{\frac {1+\left[1-{\frac {\theta ^{2}}{2}}+\mathrm {O} \left(\theta ^{3}\right)\right]}{1-\left[1-{\frac {\theta ^{2}}{2}}+\mathrm {O} \left(\theta ^{3}\right)\right]}}{\frac {x^{2}}{N^{2}}}\right\}\\&=2x\prod _{k=1}^{\frac {N-1}{2}}\left[1+{\frac {2-{\frac {\theta ^{2}}{2}}+\mathrm {O} \left(\theta ^{3}\right)}{{\frac {\theta ^{2}}{2}}+\mathrm {O} \left(\theta ^{3}\right)}}{\frac {x^{2}}{N^{2}}}\right]\\&=2x\prod _{k=1}^{\frac {N-1}{2}}\left(1+{\frac {\left(4-\theta ^{2}+\mathrm {O} \left(\theta ^{3}\right)\right)x^{2}}{\left(\theta ^{2}+\mathrm {O} \left(\theta ^{3}\right)\right)N^{2}}}\right)\\&=2x\prod _{k=1}^{\frac {N-1}{2}}\left(1+{\frac {\left(4-\left({\frac {2k\pi }{N}}\right)^{2}+\mathrm {O} \left(\left({\frac {2k\pi }{N}}\right)^{3}\right)\right)x^{2}}{\left(\left({\frac {2k\pi }{N}}\right)^{2}+\mathrm {O} \left(\left({\frac {2k\pi }{N}}\right)^{3}\right)\right)N^{2}}}\right)\\&=2x\prod _{k=1}^{\frac {N-1}{2}}\left(1+{\frac {\left(4-\left({\frac {2k\pi }{N}}\right)^{2}+\mathrm {O} \left(\left({\frac {2k\pi }{N}}\right)^{3}\right)\right)x^{2}}{(2k\pi )^{2}+\mathrm {O} \left({\frac {(2k\pi )^{3}}{N}}\right)}}\right)\end{aligned}}}
lim
N
→
∞
(
1
+
x
N
)
N
−
(
1
−
x
N
)
N
=
e
x
−
e
−
x
{\displaystyle \lim _{N\to \infty }(1+{\frac {x}{N}})^{N}-(1-{\frac {x}{N}})^{N}=e^{x}-e^{-x}}
e
x
−
e
−
x
=
2
x
∏
k
=
1
∞
(
1
+
(
4
+
o
(
1
)
)
x
2
(
2
k
π
)
2
+
o
(
1
)
)
=
2
x
∏
k
=
1
∞
(
1
+
(
1
+
o
(
1
)
)
x
2
k
2
π
2
+
o
(
1
)
)
=
2
x
∏
k
=
1
∞
(
1
+
x
2
k
2
π
2
)
{\displaystyle {\begin{aligned}e^{x}-e^{-x}&=2x\prod _{k=1}^{\infty }\left({1+{\frac {(4+o(1)){x^{2}}}{{{(2k\pi )}^{2}}+o(1)}}}\right)\ \\&=2x\prod _{k=1}^{\infty }\left({1+{\frac {(1+o(1)){x^{2}}}{{k^{2}}{\pi ^{2}}+o(1)}}}\right)\ \\&=2x\prod _{k=1}^{\infty }\left({1+{\frac {x^{2}}{{k^{2}}{\pi ^{2}}}}}\right)\ \\\end{aligned}}}
e
x
−
e
−
x
=
−
∑
k
=
0
∞
(
−
x
)
k
−
x
k
k
!
{\displaystyle e^{x}-e^{-x}=-\sum _{k=0}^{\infty }{\frac {(-x)^{k}-x^{k}}{k!}}}
k
=
3
{\displaystyle k=3}
2
x
3
3
!
{\displaystyle {\frac {2x^{3}}{3!}}}
同时展开右式可得右式的三次项为
(
∑
k
=
1
∞
2
k
2
π
2
)
x
3
{\displaystyle \left(\sum _{k=1}^{\infty }{\frac {2}{k^{2}\pi ^{2}}}\right)x^{3}}
2
3
!
=
∑
k
=
1
∞
2
k
2
π
2
{\displaystyle {\frac {2}{3!}}=\sum _{k=1}^{\infty }{\frac {2}{k^{2}\pi ^{2}}}}
π
2
6
=
∑
k
=
1
∞
1
k
2
=
ζ
(
2
)
{\displaystyle {\frac {\pi ^{2}}{6}}=\sum _{k=1}^{\infty }{\frac {1}{k^{2}}}=\zeta \left(2\right)}
Close
欧拉在1737年还发现了欧拉乘积公式 ：
∑
n
=
1
∞
1
n
s
=
∏
p
(
1
−
1
p
s
)
−
1
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\prod _{p}(1-{\frac {1}{p^{s}}})^{-1}}
证明黎曼ζ函数的欧拉乘积公式 中看到。
Re
(
s
)
>
1
{\displaystyle {\begin{smallmatrix}\operatorname {Re} (s)>1\end{smallmatrix}}}
ζ
(
s
)
>
0
{\displaystyle {\begin{smallmatrix}\zeta (s)>0\end{smallmatrix}}}
1749年，欧拉通过大胆的计算发现了(以下公式当中存在定义域谬误，后由黎曼透过解析延拓证明以下公式只适用于 Re(s ) > 1 [ 5]
ζ
(
−
1
)
=
1
+
2
+
3
+
4
+
5
+
.
.
.
=
−
1
12
{\displaystyle \zeta (-1)=1+2+3+4+5+...=-{\frac {1}{12}}}
ζ
(
−
2
)
=
1
2
+
2
2
+
3
2
+
4
2
+
5
2
+
.
.
.
=
0
{\displaystyle \zeta (-2)=1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+...=0}
ζ
(
−
3
)
=
1
3
+
2
3
+
3
3
+
4
3
+
5
3
+
.
.
.
=
1
120
{\displaystyle \zeta (-3)=1^{3}+2^{3}+3^{3}+4^{3}+5^{3}+...={\frac {1}{120}}}
1900年，希尔伯特 在巴黎的第二届国际数学家大会上作了题为《数学问题》的演讲，提出了23道最重要的数学问题，黎曼假设在其中作为第8题出现。希尔伯特－波利亚猜想 ，具体时间及场合未知。
虚部介于0与T的零点数量（蓝点）与黎曼-冯·曼格尔特公式（红线）的图像 1914年，哈那德·玻尔 和爱德蒙·兰道 证明了玻尔-兰道定理 ：含有临界线的任意带状区域都几乎包含了ζ的所有非平凡零点，表明了临界线为零点汇聚的“中心位置”。
1921年，哈代 和李特尔伍德 证明了存在常数T，使临界线上虚部位于0与T之间的非平凡零点的数量至少为
K
T
{\displaystyle {\begin{smallmatrix}KT\end{smallmatrix}}}
ζ函数与数论函数存在的联系可以通过佩龙公式 转化为它和数论函数的求和的关系：设
G
(
s
)
=
∑
n
=
1
∞
g
(
n
)
{\displaystyle G(s)={\sum _{n=1}^{\infty }}g(n)}
则由佩龙公式，
A
(
x
)
=
∑
n
≤
x
′
g
(
n
)
=
1
2
π
i
∫
c
−
i
∞
c
+
i
∞
G
(
z
)
x
z
z
d
z
{\displaystyle A(x)={\sum _{n\leq x}}'g(n)={\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }G(z){\frac {x^{z}}{z}}\,\mathrm {d} z}
其中右上角的'表示如果x是整数，那么求和的最后一项要乘以
1
2
{\displaystyle {\begin{smallmatrix}{\frac {1}{2}}\end{smallmatrix}}}
此函数和素数 的关系已由欧拉 所揭示：
ζ
(
s
)
=
∏
p
1
1
−
p
−
s
{\displaystyle \zeta (s)=\prod _{p}{\frac {1}{1-p^{-s}}}}
这是一个延展到所有的素数p 的无穷乘积 ，被称为欧拉乘积 。这是几何级数 的公式和算术基本定理 的一个结果。
log
ζ
(
s
)
=
∑
n
=
1
∞
1
n
∑
p
p
−
n
s
{\displaystyle \log \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n}}\sum _{p}p^{-ns}}
More information 首先留意到 Close
lim
ε
→
0
ζ
(
1
+
ε
)
+
ζ
(
1
−
ε
)
2
=
γ
{\displaystyle \lim _{\varepsilon \to 0}{\frac {\zeta (1+\varepsilon )+\zeta (1-\varepsilon )}{2}}=\gamma }
其中γ是欧拉-马歇罗尼常数 =
0.577215...
{\displaystyle 0.577215...}
同样由欧拉发现，ζ函数在负整数点的值是有理数，这在模形式中发挥着重要作用，而且ζ函数在负偶整数点的值为零。
事实上
ζ
(
−
n
)
=
−
B
n
+
1
n
+
1
{\displaystyle \zeta (-n)=-{\frac {B_{n+1}}{n+1}}}
B n 白努利数 。
因为 B 2n +1 =0，故ζ函数在负偶整数点的值为零。
ζ
(
x
+
i
y
)
=
∑
k
=
1
∞
cos
(
y
ln
k
)
−
i
sin
(
y
ln
k
)
k
x
,
y
∈
R
{\displaystyle \zeta (x+{\rm {i}}y)=\sum _{k=1}^{\infty }{\frac {\cos(y\ln k)-{\rm {i}}\sin(y\ln k)}{k^{x}}},y\in {\mathbb {R} }}
arg
[
ζ
(
x
+
i
y
)
]
=
−
arctan
∑
k
=
1
∞
sin
(
y
ln
k
)
k
x
∑
k
=
1
∞
cos
(
y
ln
k
)
k
x
{\displaystyle \arg[\zeta (x+{\rm {i}}y)]=-\arctan {\frac {\sum _{k=1}^{\infty }{\frac {\sin(y\ln k)}{k^{x}}}}{\sum _{k=1}^{\infty }{\frac {\cos(y\ln k)}{k^{x}}}}}}
ζ
(
−
2
n
n
∈
Z
+
)
=
0
{\displaystyle \zeta (-2n_{n\in \mathbb {Z} ^{+}})=0}
ζ
(
−
9
)
=
−
1
132
(
R
)
{\displaystyle \zeta (-9)=-{\frac {1}{132}}({\mathfrak {R}})}
ζ
(
−
7
)
=
1
240
(
R
)
{\displaystyle \zeta (-7)={\frac {1}{240}}({\mathfrak {R}})}
ζ
(
−
5
)
=
−
1
252
(
R
)
{\displaystyle \zeta (-5)=-{\frac {1}{252}}({\mathfrak {R}})}
ζ
(
−
3
)
=
1
120
(
R
)
{\displaystyle \zeta (-3)={\frac {1}{120}}({\mathfrak {R}})}
ζ
(
−
1
)
=
−
1
12
(
R
)
{\displaystyle \zeta (-1)=-{\frac {1}{12}}({\mathfrak {R}})}
ζ
(
0
)
=
−
1
2
{\displaystyle \zeta (0)=-{\frac {1}{2}}}
ζ
(
1
−
)
=
−
∞
{\displaystyle \zeta (1^{-})=-\infty }
ζ
(
1
+
)
=
∞
{\displaystyle \zeta (1^{+})=\infty }
ζ
(
2
)
=
π
2
6
{\displaystyle \zeta (2)={\frac {\pi ^{2}}{6}}}
ζ
(
4
)
=
π
4
90
{\displaystyle \zeta (4)={\frac {\pi ^{4}}{90}}}
ζ
(
6
)
=
π
6
945
{\displaystyle \zeta (6)={\frac {\pi ^{6}}{945}}}
ζ
(
8
)
=
π
8
9450
{\displaystyle \zeta (8)={\frac {\pi ^{8}}{9450}}}
ζ
(
10
)
=
π
10
93555
{\displaystyle \zeta (10)={\frac {\pi ^{10}}{93555}}}
临界线上的数值计算可以通过黎曼-西格尔公式 完成。林德勒夫猜想
ϵ
>
0
{\displaystyle {\begin{smallmatrix}\epsilon >0\end{smallmatrix}}}
ζ
(
1
2
+
i
t
)
=
O
(
t
ϵ
)
{\displaystyle \zeta \left({\frac {1}{2}}+it\right)=\mathrm {O} (t^{\epsilon })}
存档副本 . [2015-01-17 ] . （原始内容 存档于2015-03-20）. Tom M.Apostol. Introduction to Analytic Number Theory. 北京: 世界图书出版公司北京公司. 2012年1月: 55 – 56. ISBN 978-7-5100-4062-7（英语） . Theorem 3.2 If x ≥ 1 we have : C. S. Ogilvy & J. T. Anderson Excursions in Number Theory , pp. 29–35, Dover Publications Inc., 1988 ISBN 978-0-486-25778-5 
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![{\displaystyle {\begin{aligned}\left(1+{\frac {x}{N}}\right)^{N}-\left(1-{\frac {x}{N}}\right)^{N}&=\left[\left(1+{\frac {x}{N}}\right)-\left(1-{\frac {x}{N}}\right)\right]\prod _{k=1}^{\frac {N-1}{2}}\left[\left(1+{\frac {x}{N}}\right)^{2}-2\left(1+{\frac {x}{N}}\right)\left(1-{\frac {x}{N}}\right)\cos \left({\frac {2\pi k}{N}}\right)+\left(1-{\frac {x}{N}}\right)^{2}\right]\\&={\frac {2x}{N}}\prod _{k=1}^{\frac {N-1}{2}}\left[2+{\frac {2x^{2}}{N^{2}}}-2\left(1-{\frac {x^{2}}{N^{2}}}\right)\cos \left({\frac {2\pi k}{N}}\right)\right]\\&={\frac {2x}{N}}\prod _{k=1}^{\frac {N-1}{2}}\left[{2+{\frac {2{x^{2}}}{N^{2}}}-2\cos \left({\frac {2\pi k}{N}}\right)+{\frac {2{x^{2}}}{N^{2}}}\cos({\frac {2\pi k}{N}})}\right]\\&={\frac {4x}{N}}\prod _{k=1}^{\frac {N-1}{2}}\left({(1-\cos({\frac {2\pi k}{N}}))+(1+\cos({\frac {2\pi k}{N}})){\frac {x^{2}}{N^{2}}}}\right)\\&={\frac {4x}{N}}\prod _{k=1}^{\frac {N-1}{2}}\left\{\left[1-\cos \left({\frac {2\pi k}{N}}\right)\right]\left[{1+{\frac {1+\cos \left({\frac {2\pi k}{N}}\right)}{1-\cos({\frac {2\pi k}{N}})}}{\frac {x^{2}}{N^{2}}}}\right]\right\}\\\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/23ed868ba0fd3d14af72b827080bb13dcab2a8e8)
![{\displaystyle {\begin{aligned}\left(1+{\frac {x}{N}}\right)^{N}-\left(1-{\frac {x}{N}}\right)^{N}&=2x\prod _{k=1}^{\frac {N-1}{2}}\left[1+{\frac {1+\cos \left({\frac {2\pi k}{N}}\right)}{1-\cos \left({\frac {2\pi k}{N}}\right)}}{\frac {x^{2}}{N^{2}}}\right]\\&=2x\prod _{k=1}^{\frac {N-1}{2}}\left\{1+{\frac {1+\left[1-{\frac {\theta ^{2}}{2}}+\mathrm {O} \left(\theta ^{3}\right)\right]}{1-\left[1-{\frac {\theta ^{2}}{2}}+\mathrm {O} \left(\theta ^{3}\right)\right]}}{\frac {x^{2}}{N^{2}}}\right\}\\&=2x\prod _{k=1}^{\frac {N-1}{2}}\left[1+{\frac {2-{\frac {\theta ^{2}}{2}}+\mathrm {O} \left(\theta ^{3}\right)}{{\frac {\theta ^{2}}{2}}+\mathrm {O} \left(\theta ^{3}\right)}}{\frac {x^{2}}{N^{2}}}\right]\\&=2x\prod _{k=1}^{\frac {N-1}{2}}\left(1+{\frac {\left(4-\theta ^{2}+\mathrm {O} \left(\theta ^{3}\right)\right)x^{2}}{\left(\theta ^{2}+\mathrm {O} \left(\theta ^{3}\right)\right)N^{2}}}\right)\\&=2x\prod _{k=1}^{\frac {N-1}{2}}\left(1+{\frac {\left(4-\left({\frac {2k\pi }{N}}\right)^{2}+\mathrm {O} \left(\left({\frac {2k\pi }{N}}\right)^{3}\right)\right)x^{2}}{\left(\left({\frac {2k\pi }{N}}\right)^{2}+\mathrm {O} \left(\left({\frac {2k\pi }{N}}\right)^{3}\right)\right)N^{2}}}\right)\\&=2x\prod _{k=1}^{\frac {N-1}{2}}\left(1+{\frac {\left(4-\left({\frac {2k\pi }{N}}\right)^{2}+\mathrm {O} \left(\left({\frac {2k\pi }{N}}\right)^{3}\right)\right)x^{2}}{(2k\pi )^{2}+\mathrm {O} \left({\frac {(2k\pi )^{3}}{N}}\right)}}\right)\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/a3463889e69fa6c26e55d810d8a2ef0a8dc7c440)
, 
...
![{\displaystyle \arg[\zeta (x+{\rm {i}}y)]=-\arctan {\frac {\sum _{k=1}^{\infty }{\frac {\sin(y\ln k)}{k^{x}}}}{\sum _{k=1}^{\infty }{\frac {\cos(y\ln k)}{k^{x}}}}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/e900016d87e7c6f5d1672b28108042d891a05718)
