Fonksyon zeta Riemann

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Se yon fonksyon matematik rele pa matematisyen alman Bernhard Riemann.

Definsiyon[edite | modifier le wikitexte]

Li defini pa seri sa :


\zeta(s) =
\sum_{n=1}^\infty \frac{1}{n^s}
\!


Euler[edite | modifier le wikitexte]

Lyen ki egziste ant fonksyon zeta ak nonm premye yo te dekouvè pa matematisyen Leonhard Euler :


\begin{align}
\zeta(s)=\sum_{n\geq 1}\frac{1}{n^s}& = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}} \\
& = \left(1 + \frac{1}{2^s} + \frac{1}{4^s} + \cdots \right) \left(1 + \frac{1}{3^s} + \frac{1}{9^s} + \cdots \right) \cdots \left(1 + \frac{1}{p^s} + \frac{1}{p^{2s}} + \cdots \right) \cdots,
\end{align}
\!

p ap deziyen yon nonm premye.Seri sa genyen yon sans lè pati reyèl Re(s) > 1.

Propryete[edite | modifier le wikitexte]

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Kèk valè pou konnen[edite | modifier le wikitexte]

\zeta(0) = -1/2,\!
\zeta(1/2) \approx -1.4603545088095868,\!
\zeta(1) = 1 + \frac{1}{2} + \frac{1}{3} + \cdots = \infty;\! Seri amonik.
\zeta(3/2) \approx 2.612;\! itilize nan kalkil tanperati kritik pou kondansa Bose–Einstein nan domèn fizik
\zeta(2) = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots = \frac{\pi^2}{6} \approx 1.645;\! demonstrasyon egalite sa rele tou pwoblèm Basel.
\zeta(5/2) \approx 1.341.\!
\zeta(3) = 1 + \frac{1}{2^3} + \frac{1}{3^3} + \cdots \approx 1.202;\! relekonstant Apéry.
\zeta(7/2) \approx 1.127\!
\zeta(4) = 1 + \frac{1}{2^4} + \frac{1}{3^4} + \cdots = \frac{\pi^4}{90} \approx 1.0823;\! lwa Stefan–Boltzmann epi apwoksimasyon Wien nan domèn fizik.

Fonksyon - analiz[edite | modifier le wikitexte]

Fonksyon zeta an satisfèt ekwasyon fonksyonèl sa tou :


\zeta(s) = 2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s)
\!


Lyen[edite | modifier le wikitexte]