Fonksyon zeta Riemann

• 
  Fonksyon zeta Riemann ζ(s) nan [[plan konplèks]. La couleur d'un point s code la valeur de ζ(s) : 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 Re(s) = 1/2 (droite verticale) sont les zéros.]]

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

Definsiyon

Li defini pa seri sa :

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

• 
  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 :

${\begin{aligned}\zeta (s)=\sum _{n\geq 1}{\frac {1}{n^{s}}}&=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}}\\&=\left(1+{\frac {1}{2^{s}}}+{\frac {1}{4^{s}}}+\cdots \right)\left(1+{\frac {1}{3^{s}}}+{\frac {1}{9^{s}}}+\cdots \right)\cdots \left(1+{\frac {1}{p^{s}}}+{\frac {1}{p^{2s}}}+\cdots \right)\cdots ,\end{aligned}}\!$

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

Pwopryete

Kèk valè pou konnen

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

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

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

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

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