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

${\displaystyle \left|\pi (x)-\int _{0}^{x}{\frac {\mathrm {d} t}{\ln(t)}}\right|={\mathcal {O}}(x^{1/2+\varepsilon }),}$

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 :

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

• 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.

• 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

• Clay Mathematics Institute, 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", 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, 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
• Zetagrid (2002) A distributed computing project that attempted to disprove Riemann's hypothesis; closed in November 2005
• 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., 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, 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, 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, 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

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

• The Millennium Grand Challenge in Mathematics
• The Millennium Prize Problems