기호 ζ

ζ(x) = 1 + 2^^−x^^ + 3^^−x^^ + 4^^−x^^ + ⋯

$\zeta(x)=\sum_{n=1}^{\infty}\frac1{n^x}$

$\zeta(s)=\frac1{1^s}+\frac1{2^s}+\frac1{3^s}+\frac1{4^s}+\frac1{5^s}+\cdots$

$\zeta(s)=\sum_{n=1}^{\infty}n^{-s}$

[[무한급수,infinite_series]]로 나타남

ζ(1)인 경우는 [[조화급수,harmonic_series]]의 일종.
관련: [[p급수,p-series]]

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Euler Product Formula
 $\zeta(s)=\sum_{n}\frac{1}{n^s}=\prod_{p}\frac1{1-p^{-s}}$
n, p는 양수이고 p는 [[소수,prime_number]]

// tmp via https://mathphysics.tistory.com/703 '리만제타함수의 기본성질' slide
{
[[오일러_곱,Euler_product]]? ...[[곱,product]]

Euler product:
 $\zeta(s)=\prod_p \frac1{1-p^{-s}} \;\; (\operatorname{Re}(s)>1)$
그럼 저 위에 Re(s)>1 ...조건 필요? 

chk.
}
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[[제타함수,zeta_function]]의 정의역을 복소수로 확장한([[해석적연속,analytic_continuation]]) 것이 리만의 제타함수.
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이것의 일반화로 Hurwitz_zeta_function ..

http://specialfunctionswiki.org/index.php/Hurwitz_zeta
[[WpEn:Hurwitz_zeta_function]]
[[WpKo:후르비츠_제타_함수]]

[[MathNote:후르비츠_제타함수(Hurwitz_zeta_function)]]
... Ggl:"Hurwitz zeta function"
----
Up: [[함수,function]] > [[제타함수,zeta_function]]

[[https://oeis.org/wiki/Riemann_ζ_function]]
[[WpEn:Riemann_zeta_function]]
[[WpKo:리만_제타_함수]]
https://mathworld.wolfram.com/RiemannZetaFunction.html
https://www.proofwiki.org/wiki/Definition:Riemann_Zeta_Function

https://artofproblemsolving.com/wiki/index.php/Riemann_zeta_function

zeta fn.

https://mathworld.wolfram.com/ZetaFunction.html
https://encyclopediaofmath.org/wiki/Zeta-function

tmp
https://everything2.com/title/Riemann+zeta+function
{
//LionMan
infinite_sum 으로 처음 정의
 $\operatorname{Zeta}(n):=\sum_{m=0}^{\infty} \frac1{m^n}$
[[복소평면,complex_plane]]으로 해석적 확장하면 (analytically extended) ([[해석적연속,analytic_continuation]])
 $\operatorname{Zeta}(z):=\frac{\int_{0}^{\infty}\frac{u^{z-1}}{e^u-1}du}{\Gamma(z)}$
 $\operatorname{Zeta}(z):=\frac{\textstyle\int_{0}^{\infty}\frac{u^{z-1}}{e^u-1}du}{\Gamma(z)}$

//abiessu
오일러가 증명하길
 $\operatorname{Zeta}(z):=\sum_{m=0}^{\infty}\frac1{m^z}$
는 다음과 같다. $\mathbb{P}$ 는 [[소수,prime_number]].
 $\prod_{p\in\mathbb{P}}\frac1{1-p^{-z}}$
다시 말해 다음과 같다.
 $\frac1{1-2^{-z}} \cdot \frac1{1-3^{-z}} \cdot \frac1{1-5^{-z}} \cdot \frac1{1-7^{-z}} \cdot \cdots$
....tbw
}

MKL
[[리만_가설,Riemann_hypothesis]]

http://specialfunctionswiki.org/index.php/Riemann_zeta
Libre:리만_제타함수