Fonksyon zeta Riemann

Li defini pa seri sa :

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

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  Fonksyon zeta Riemann zeta pou reyèl s > 1

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

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

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

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

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

${\displaystyle \zeta (1)=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots =\infty ;\!}$ seri amonik.

${\displaystyle \zeta (3/2)\approx 2.612;\!}$ itilize nan kalkil tanperati kritik pou kondansa Bose–Einstein nan domèn fizik

${\displaystyle \zeta (2)=1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}\approx 1.645;\!}$ demonstrasyon egalite sa rele tou pwoblèm Basel.

${\displaystyle \zeta (5/2)\approx 1.341.\!}$

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

${\displaystyle \zeta (7/2)\approx 1.127\!}$

${\displaystyle \zeta (4)=1+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+\cdots ={\frac {\pi ^{4}}{90}}\approx 1.0823;\!}$ lwa Stefan–Boltzmann epi apwoksimasyon Wien nan domèn fizik.

Fonksyon zeta an satisfèt ekwasyon fonksyonèl sa tou :

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

• 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