Fonksyon zeta Riemann

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  Fonksyon zeta Riemann ζ(s) nan [[plan konplèks]. La couleur d'un point s code la valeur de Modèl:Math : des couleurs vives indiquent des valeurs proches de 0 et la nuance indique l'argument de la valeur. Le point blanc pour s = 1 est le pôle ; les points noirs sur l'axe réel négatif (demi-droite horizontale) et sur la droite critique Modèl:Math = Modèl:Sfrac (droite verticale) sont les zéros.]]

Fonksyon zeta Riemann se yon fonksyon matematik rele pa matematisyen alman Bernhard Riemann.

Definsiyon

Li defini pa seri sa :

$\zeta (s)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}$

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  Fonksyon zeta Riemann zeta pou reyèl s > 1

Euler

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

$\begin{array}{cc}\zeta (s)=\sum _{n\ge 1}\frac{1}{n^s}& =\prod _{p\text{ prime}}\frac{1}{1-p^{-s}}\\ & =(1+\frac{1}{2^s}+\frac{1}{4^s}+\cdots )(1+\frac{1}{3^s}+\frac{1}{9^s}+\cdots )\cdots (1+\frac{1}{p^s}+\frac{1}{p^{2s}}+\cdots )\cdots ,\end{array}$

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

Pwopryete

Kèk valè pou konnen

$\zeta (0)=-1/2,$

$\zeta (1/2)\approx -1.4603545088095868,$

$\zeta (1)=1+\frac{1}{2}+\frac{1}{3}+\cdots =\mathrm{\infty };$ 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.

$\zeta (5/2)\approx 1.341.$

$\zeta (3)=1+\frac{1}{2^3}+\frac{1}{3^3}+\cdots \approx 1.202;$ rele konstant 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

Fonksyon zeta an satisfèt ekwasyon fonksyonèl sa tou :

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

Lyen deyò

• Riemann Zeta Function, in Wolfram Mathworld - an explanation with a more mathematical approach
• Tables of selected zeroes
• File with 1,000,000 zeros and accurate to about 60+ digits (To download compressed archive, click on Download Now... button.)
• Prime Numbers Get Hitched A general, non-technical description of the significance of the zeta function in relation to prime numbers.
• X-Ray of the Zeta Function Visually-oriented investigation of where zeta is real or purely imaginary. (PDF)
• Formulas and identities for the Riemann Zeta function functions.wolfram.com
• Riemann Zeta Function and Other Sums of Reciprocal Powers, section 23.2 of Abramowitz and Stegun