Ipotèz Riemann

Se yon ipotèz matematik, yon konjekti fèt pa Bernhard Riemann.

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| 

## Referans
### Referans istorik
* Bernhard Riemann, *[https://web.archive.org/web/20130523061451/http://www.claymath.org/millennium/Riemann_Hypothesis/1859_manuscript/ 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
*  *(A relationship in terms of Harmonic numbers.)*
* *(no author credited)*, *[http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeroscompute.html 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

### Referans popilè
* Clay Mathematics Institute, *[https://web.archive.org/web/20080108223641/http://www.claymath.org/millennium/ Millennium Problems]*, (2000) (Announcement of the million dollar rewards for solutions to famous problems in mathematics)
* Marcus du Sautoy, *The Music of the Primes*, HarperCollins, 2003
* Marcus du Sautoy, "Prime Numbers Get Hitched", *[https://web.archive.org/web/20200606033642/http://www.seedmagazine.com/ Seed Magazine]*" (03/27/2006)
* Daniel Rockmore, *Stalking the Riemann Hypothesis : The Quest to Find the Hidden Law of Prime Numbers*, Pantheon Books, New York, 2005, ISBN 0-375-42136-X
* John Derbyshire, *[http://www.nap.edu/catalog/10532.html Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics]*, Joseph Henry Press (April 23, 2003), ISBN 0-309-08549-7. 448 page book at a non-specialist level, can be read online for free
* *[https://web.archive.org/web/20131005173705/http://www.zetagrid.net/ Zetagrid]* (2002) A distributed computing project that attempted to disprove Riemann's hypothesis; closed in November 2005
* *[[A Beautiful Mind (film)|A Beautiful Mind]]* (2001, produced by Ron Howard). In the film, the mathematician John Nash attempts to prove the Riemann Hypothesis
* *[[Law & Order: Criminal Intent]]:* In season 2, episode 2, "Bright Boy", the detective played by Vincent D'Onofrio tricks a father into confessing to a murder by telling him his son has solved the Riemann Hypothesis
* *[[Numb3rs]]*, (CBS), season 1 episode 5, "Prime Suspect", Criminals kidnap a mathematician's daughter and demand his allegedly complete proof of Riemann's Hypothesis as ransom
* Ed Pegg, Jr., *[https://web.archive.org/web/20041102173644/http://www.maa.org/editorial/mathgames/mathgames_10_18_04.html Ten Trillion Zeta Zeros]*, (2004) Math Games website. A discussion of Xavier Gourdon's calculation of the first ten trillion non-trivial zeros
* de Vries, *[http://math-it.org/Mathematik/Riemann/RiemannApplet.html The Graph of the Riemann Zeta function ζ(s)]* (2004), a simple animated Java applet
* Erica Klarreich, "Prime Time", *[[New Scientist]]* - November 11, 2000, p. 32. A simple introduction to the Riemann Hypothesis, and its connection to prime numbers and quantum systems
* QEDen, *[https://web.archive.org/web/20070926223705/http://www.qeden.com/wiki/The_Riemann_Hypothesis A wiki dedicated to solving the millennium problems]*
* Perplex City, an Alternate Reality Game soliciting answers to various puzzles from its players, is offering 60 points to any player to submit a proof of the Riemann Hypothesis on card 238 of Season 1, entitled "Riemann"
* *[[Proof (2005 film)|Proof]]*, a film starring Gwynneth Paltrow and Anthony Hopkins, in which a character is purported to have solved a problem that sounds like the Riemann Hypothesis
* *[[PopCo]]*, a novel wherein the main character's grandmother used to spend all her free time trying to prove the Riemann Hypothesis
* Thomas Pynchon, the novel *Against The Day*, wherein the Riemann Hypothesis and the question of non-trivial zeros are recurring tropes and preoccupations of the students and spies gathered at Göttingen University in the years preceding World War I
* Karl Sabbagh, *The Riemann Hypothesis: the greatest unsolved problem in mathematics*, (2003) Farrar, Straus and Giroux, ISBN 0-374-25007-3. Also (2004) First American paperback edition. Conversations with mathematicians working on the problem. The reader should not expect to learn anything about the problem itself

### Lòt referans
* Bollobas, Bela, foreword to Littlewood's Miscellany, Cambridge University Press, 1986

## Lyen deyò
* The Millennium Grand Challenge in Mathematics
* The Millennium Prize Problems