# Ipotèz Riemann

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Se yon ipotèz matematik, yon konjekti fèt pa Bernhard Riemann. pati reyèl an wouj, pati imajinè an ble de fonksyon Zeta Riemann anlè liy kritik Re(s) = 1/2

Kesyon l poze ap prezante konsa : èske tout zero fonksyon Zeta Riemann an genyen 1/2 pou pati reyèl ?

Li twouve yon fomilasyon pli senp pa :

Pou tout ε > 0, nou genyen

$\left|\pi (x)-\int _{0}^{x}{\frac {\mathrm {d} t}{\ln(t)}}\right|={\mathcal {O}}(x^{1/2+\varepsilon }),$ lè π(x) ap deziyen kantite nonm premye ki anvan x, ln(x) se fonksyon logaritm, evalye nan x, epi a dwat ou ap twouve notasyon Landau.

Yon vèsyon ki pa asenptotik pa Lowell Schoenfeld, ap di ke ipotèz Riemann an ekivalan a :

$\left|\pi (x)-\int _{0}^{x}{\frac {\mathrm {d} t}{\ln(t)}}\right|<{\frac {1}{8\pi }}{\sqrt {x}}\,\ln(x),\qquad {\text{pou tout }}x\geq 2657.$ ## Referans

### Referans istorik

• Bernhard Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, (1859) Monatsberichte der Berliner Akademie. (This site provides both a facsimile of the original manuscripts, as well as English translations.)
• Jacques Hadamard, Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques, Bulletin Société Mathématique de France 14 (1896) pp 199-220.

### Teknik jodi jou

• H. M. Edwards, Riemann's Zeta Function, Academic Press, 1974. (Reprinted by Dover Publications, 2001 ISBN 0-486-41740-9)
• E. C. Titchmarsh, The Theory of the Riemann Zeta Function, second revised (Heath-Brown) edition, Oxford University Press, 1986
• Jeffrey Lagarias (2002). « An Elementary Problem Equivalent to the Riemann Hypothesis ». American Mathematical Monthly 109: 534–543. (A relationship in terms of Harmonic numbers.)
• (no author credited), Computation of zeros of the Zeta function (2004). (Reviews the GUE hypothesis, provides an extensive bibliography as well).
• Schoenfeld, Lowell. "Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II." Mathematics of Computation 30 (1976), no. 134, 337--360.
• Conrey, J. Brian. "the Riemann Hypothesis" Notices of the American Mathematical Society, March 2003, 341-353. Available free http://www.ams.org/notices/200303/fea-conrey-web.pdf

### Lòt referans

• Bollobas, Bela, foreword to Littlewood's Miscellany, Cambridge University Press, 1986