1. Clay Mathematics Institute Article by Enrico Bombieri (PDF) and video by Jeff Vaaler (.ram) from the Clay Mathematics Institute. http://www.claymath.org/millennium/Riemann_Hypothesis/

Clay Mathematics Institute Dedicated to increasing and disseminating mathematical knowledge HOME ABOUT CMI PROGRAMS AWARDS ... PUBLICATIONS Riemann Hypothesis Some numbers have the special property that they cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. Such numbers are called prime numbers, and they play an important role, both in pure mathematics and its applications. The distribution of such prime numbers among all natural numbers does not follow any regular pattern, however the German mathematician G.F.B. Riemann (1826 - 1866) observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function s s s called the Riemann Zeta function . The Riemann hypothesis asserts that all interesting solutions of the equation lie on a certain vertical straight line. This has been checked for the first 1,500,000,000 solutions. A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of prime numbers. The Millennium Problems Lecture by Jeff Vaaler at the University of Texas (video) Facsimile of Riemann's 1859 manuscript Return to top Contact Search Clay Mathematics Institute

2. Riemann Hypothesis - Wikipedia, The Free Encyclopedia In mathematics, the Riemann hypothesis, proposed by Bernhard Riemann (1859), is a conjecture about the distribution of the zeros of the Riemann zeta http://en.wikipedia.org/wiki/Riemann_hypothesis

Riemann hypothesis From Wikipedia, the free encyclopedia Jump to: navigation search The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re( s ) = 1/2. The first non-trivial zeros can be seen at Im( s ) = ±14.135, ±21.022 and ±25.011. Millennium Prize Problems P versus NP problem Hodge conjecture Poincaré conjecture ... solution Riemann hypothesis Yang–Mills existence and mass gap Navier–Stokes existence and smoothness Birch and Swinnerton-Dyer conjecture v ... e In mathematics, the Riemann hypothesis , proposed by Bernhard Riemann ), is a conjecture about the distribution of the zeros of the Riemann zeta function which states that all non-trivial zeros of the Riemann zeta function have real part 1/2. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields The Riemann hypothesis implies results about the distribution of prime numbers that are in some ways as good as possible. Along with suitable generalizations, it is considered by some mathematicians to be the most important unresolved problem in pure mathematics Bombieri 2000 The Riemann zeta function ζ( s ) is defined for all complex numbers s s trivial zeros . The Riemann hypothesis is concerned with the non-trivial zeros, and states that:

3. Riemann Hypothesis The most important open question in number theory and, possibly, in the whole of mathematics. David Hilbert said If I were to awaken after having slept for a thousand years http://www.daviddarling.info/encyclopedia/R/Riemann_hypothesis.html

NUMBER THEORY A B C ... CONTACT entire Web this site Riemann hypothesis The most important open question in number theory and, possibly, in the whole of mathematics. David Hilbert said: "If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?" A $1 million prize has been offered by the Clay Mathematics Institute for a proof. The hypothesis was first formulated by Bernhard Riemann in 1859, was included in David Hilbert's list of challenging problems for 20th-century mathematicians, and is widely believed to be true. Yet a proof remains tantalizingly out of reach. What the Riemann hypothesis says is that the non-trivial zeros of the Riemann zeta function all have real part equal to 1/2. Broadly speaking, the hypothesis asserts that there is an underlying order, akin to musical harmonics, in the way prime numbers are distributed. It's known that for any given number n there are approximately n /log n prime numbers that are less than n . The formula is not exact: sometimes it is a little high and sometimes a little low. Riemann looked at these deviations and found that they contain periodicities. His hypothesis quantifies and formalizes this discovery, positing that the zeros of the zeta function can be regarded as the harmonic frequencies in the distribution of primes. If the Riemann hypothesis turns out to be true, what do these harmonics in the "music" of the primes mean?

4. Riemann Hypothesis - AoPSWiki Apr 10, 2009 The Riemann Hypothesis is a wellknown conjecture in analytic number theory that states that all nontrivial zeros of the Riemann zeta http://www.artofproblemsolving.com/Wiki/index.php/Riemann_Hypothesis

Art of Problem Solving LOGIN/REGISTER Home Bookstore AoPS Curriculum ... Articles You are here: Resources AoPSWiki Search Wiki Page Tools Page Discussion View source History Personal tools Talk for this IP Log in Navigation Main Page Community portal Current events Recent changes ... Help Toolbox What links here Related changes Special pages Printable version ... Permanent link Riemann Hypothesis From AoPSWiki The Riemann Hypothesis is a well-known conjecture in analytic number theory that states that all nontrivial zeros of the Riemann zeta function have real part . From the functional equation for the zeta function, it is easy to see that when . These are called the trivial zeros. This hypothesis is one of the seven millenium questions The Riemann Hypothesis is an important problem in the study of prime numbers . Let denote the number of primes less than or equal to x , and let . Then an equivalent statement of the Riemann hypothesis is that One fairly obvious method to prove the Riemann Hypothesis is to consider the reciprocal of the zeta function, , where refers to the Möbius function . Then one might try to show that admits an analytic continuation to . Let be the Mertens function . It is easy to show that if for sufficiently large , then the Riemann Hypothesis would hold. The Riemann Hypothesis would also follow if

5. Riemann Hypothesis - Uncyclopedia, The Content-free Encyclopedia Sep 30, 2009 The Riemann Hypothesis was a key mathematical conjecture in the field of theoretical gangsta rap which remained unsolved for centuries. http://uncyclopedia.wikia.com/wiki/Riemann_Hypothesis

6. Riemann Hypothesis - Conservapedia This page was last modified on 15 August 2010, at 1801. This page has been accessed 1,758 times. Privacy policy; About Conservapedia; Disclaimers http://www.conservapedia.com/Riemann_hypothesis

Riemann hypothesis From Conservapedia Jump to: navigation search The Riemann hypothesis states that the non-trivial zeros of the Riemann Zeta function function all have real component . The conjecture was first proposed by Bernhard Riemann in 1859 and is considered by many to be one of the greatest unsolved problems in mathematics. The problem was listed as one of the seven Millennium problems by the Clay Mathematics Institute and there is a million dollar prize for its solution. The statement is essentially equivalent to the claim that the error term in the pime number theorem is small. Alternatively, the Riemann hypothesis can be thought of as a statement that the prime numbers are very smoothly distributed. References http://www.claymath.org/millennium/ Retrieved from " http://www.conservapedia.com/Riemann_hypothesis Category Complex analysis Views Page talk page View source History Personal tools Log in Search Popular Links Main Page Educational Index Debate Topics Recent changes ... Random page Help Help Index Statistics Conservapedia Commandments World History World History Lectures World History Homework Edit Console What links here Related changes Special pages Printable version ... Permanent link This page was last modified on 15 August 2010, at 18:01.

7. Generalized Riemann Hypothesis - Wikipedia, The Free Encyclopedia The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical http://en.wikipedia.org/wiki/Generalized_Riemann_hypothesis

Generalized Riemann hypothesis From Wikipedia, the free encyclopedia Jump to: navigation search The Riemann hypothesis is one of the most important conjectures in mathematics . It is a statement about the zeros of the Riemann zeta function . Various geometrical and arithmetical objects can be described by so-called global L-functions , which are formally similar to the Riemann zeta-function. One can then ask the same question about the zeros of these L -functions, yielding various generalizations of the Riemann hypothesis. Many mathematicians believe these generalizations of the Riemann hypothesis to be true. The only cases of these conjectures which have been proven occur in the function field case (not the number field case). Global L -functions can be associated to elliptic curves number fields (in which case they are called Dedekind zeta-functions Maass forms , and Dirichlet characters (in which case they are called Dirichlet L-functions ). When the Riemann hypothesis is formulated for Dedekind zeta-functions, it is known as the extended Riemann hypothesis (ERH) and when it is formulated for Dirichlet L -functions, it is known as the

8. Riemann Hypothesis - Simple English Wikipedia, The Free Encyclopedia The Riemann hypothesis is a mathematical conjecture. Many people think that finding a proof of the hypothesis is one of the hardest and most important unsolved problems of pure http://simple.wikipedia.org/wiki/Riemann_hypothesis

Riemann hypothesis From Wikipedia, the free encyclopedia Jump to: navigation search It has been suggested that the text on Riemann zeta function be merged into (added to) this article or section. ( Discuss The Riemann zeta function, in the complex plane. The real part ( Re( s ) of the number is drawn horizontally, the imaginary part Im( s vertically. White dots show the zeros where Re( s Click to get a full view. The Riemann hypothesis is a mathematical conjecture . Many people think that finding a proof of the hypothesis is one of the hardest and most important unsolved problems of pure mathematics The hypothesis is named after Bernhard Riemann . It is about a special function , the Riemann zeta function . This function inputs and outputs complex number values. The inputs that give the output zero are called zeros of the zeta function. Many zeros have been found. The "obvious" ones to find are the negative even integers. This follows from Riemann's functional equation. More have been computed and have real part 1/2. The hypothesis states all the undiscovered zeros must have real part 1/2. The functional equation also says all zeros (except the "obvious" ones) must be in the critical strip: real part is between and 1. The Riemann hypothesis says more: they are on the line given, in the image on the right (the white dots). If the hypothesis is false, this would mean that there are white dots which are not on the line given.

9. Mathematician Vasiliu Lucilius | Riemann Hypothesis - Solved The Riemann Hypothesis is one of the seven Millennium problems on the list of the Clay Mathematics Institute of Cambridge, Massachusetts (CMI) and its http://www.number-theory.eu/riemann_hypothesis.php

The Riemann Hypothesis About Me My Work The Riemann Hypothesis Guestbook ... Contact Mathematician Vasiliu Lucilius Five Major Theoretical Breakthroughs in the Superior Mathematics Chapter 1. Fractional powers with rational exponent of the monotonous operators of class on Chapter 2. Using the symmetry group in demonstrating the radiality of several semilinear biharmonic equations' solutions Chapter 3. A study on zeroes of the function (z) of Riemann Chapter 4. The problem of invariant Subspaces Chapter 5. A study on Navier-Stokes equations The Riemann Hypothesis is one of the seven Millennium problems on the list of the Clay Mathematics Institute of Cambridge, Massachusetts (CMI) and its solution has allocated a $1 million prize. Starting in 1995 and throughout 2000, I have focused on the Riemann Hypothesis , managing to transform a Complex Analysis problem into a Convex Functional Analysis problem. It is in my intention to underline the fact that Romanian mathematicians gave credit to world-wide mathematics. Just lately, in the year of 2002, the Romanian mathematician

10. Problems Of The Millennium The Riemann Hypothesis Problems of the Millennium the Riemann Hypothesis E. Bombieri I. The problem. The Riemannzeta function is the function of the complex variable s , defined in the halfplane 1 http://www.claymath.org/millennium/Riemann_Hypothesis/Official_Problem_Descripti

11. Riemann Hypothesis - Definition The Riemann hypothesis, first formulated by Bernhard Riemann in 1859, is a conjecture about the distribution of the zeros of the Riemann zeta function (s) . http://www.wordiq.com/definition/Riemann_hypothesis

Riemann hypothesis - Definition RH directs here. RH is also the common abbreviation for the soap Ryan's Hope The Riemann hypothesis , first formulated by Bernhard Riemann in , is a conjecture about the distribution of the zeros of the Riemann zeta function s ). The Riemann zeta function is defined for all complex numbers s s s s The real part of any non-trivial zero of the Riemann zeta function is 1/2. Thus the non-trivial zeros should lie on the so-called critical line it with t a real number and i the imaginary unit . The Riemann zeta function along the critical line is sometimes studied in terms of the Z function , whose real zeros correspond to the zeros of the zeta function on the critical line. The Riemann hypothesis is one of the most important open problems of contemporary mathematics ; a $1,000,000 prize has been offered by the Clay Mathematics Institute for a proof. Most mathematicians believe the Riemann hypothesis to be true. ( J. E. Littlewood and Atle Selberg have been reported as skeptical.) In 2004, Xavier Gourdon verified the Riemann hypothesis through the first ten trillion non-trivial zeros using the Odlyzko-Schönhage algorithm. Contents showTocToggle("show","hide")

12. Riemann Hypothesis - Uncyclopedia, The Content-free Encyclopedia The Riemann Hypothesis was a key mathematical conjecture in the field of theoretical gangsta rap which remained unsolved for centuries. It concerns the behavior of the quality http://uncyclopedia.wikia.com/wiki/Riemann_hypothesis

13. Wapedia - Wiki: Riemann Hypothesis In mathematics, the Riemann hypothesis, proposed by Bernhard Riemann (1859), is a conjecture about the distribution of the zeros of the Riemann http://wapedia.mobi/en/Riemann_hypothesis

Wiki: Riemann hypothesis In mathematics, the Riemann hypothesis , proposed by Bernhard Riemann ), is a conjecture about the distribution of the zeros of the Riemann zeta function which states that all non-trivial zeros of the Riemann zeta function have real part 1/2. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re( s ) = 1/2. The first non-trivial zeros can be seen at Im( s ) = ±14.135, ±21.022 and ±25.011. The Riemann hypothesis implies results about the distribution of prime numbers that are in some ways as good as possible. Along with suitable generalizations, it is considered by some mathematicians to be the most important unresolved problem in pure mathematics Bombieri 2000 The Riemann zeta function ζ( s ) is defined for all complex numbers s ≠ 1. It has zeros at the negative even integers (i.e. at s = −2, −4, −6, ...). These are called the trivial zeros . The Riemann hypothesis is concerned with the non-trivial zeros, and states that:

14. Riemann Hypothesis Summary And Analysis Summary | BookRags.com Riemann hypothesis summary with 15 pages of lesson plans, quotes, chapter summaries, analysis, encyclopedia entries, essays, research information, and more. http://www.bookrags.com/Riemann_hypothesis

15. Riemann Hypothesis - Discussion And Encyclopedia Article. Who Is Riemann Hypothe Riemann hypothesis. Discussion about Riemann hypothesis. Ecyclopedia or dictionary article about Riemann hypothesis. http://www.knowledgerush.com/kr/encyclopedia/Riemann_hypothesis/

16. Riemann Hypothesis - On Opentopia, Find Out More About Riemann Hypothesis In mathematics, the Riemann hypothesis (also called the Riemann zetahypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous unsolved problems. http://encycl.opentopia.com/term/Riemann_hypothesis

About Opentopia Opentopia Directory Encyclopedia ... Tools Riemann hypothesis Encyclopedia R RI RIE : Riemann hypothesis Millennium Prize Problems P versus NP The Hodge conjecture The Poincaré conjecture The Riemann hypothesis Yang–Mills existence and mass gap Navier-Stokes existence and smoothness The Birch and Swinnerton-Dyer conjecture Unsolved problems in mathematics Is the real part of a non-trivial zero of the Riemann zeta function In mathematics , the Riemann hypothesis (also called the Riemann zeta-hypothesis ), first formulated by Bernhard Riemann in 1859, is one of the most famous unsolved problems. It has been an open question for well over a century, despite attracting concentrated efforts from many outstanding mathematicians. Unlike some other celebrated problems, it is more attractive to professionals in the field than to amateurs. The Riemann hypothesis is a conjecture about the distribution of the zero s of the Riemann zeta-function s ). The Riemann zeta-function is defined for all complex number s s s s s trivial zeros . The Riemann hypothesis is concerned with the non-trivial zeros, and states that:

17. Riemann Hypothesis In Encyclopedia Riemann hypothesis in Encyclopedia in Encyclopedia The real part (red) and imaginary part (blue) of the Riemann zetafunction along the critical line Re(s) = 1/2. http://www.tutorgig.com/ed/Riemann_hypothesis

18. Online Encyclopedia And Dictionary - Riemann Hypothesis RH directs here. RH is also the common abbreviation for the soap opera Ryan's Hope. In mathematics, the Riemann hypothesis, first formulated by Bernhard Riemann in 1859, is one http://fact-archive.com/encyclopedia/Riemann_hypothesis

Search The Online Encyclopedia and Dictionary Encyclopedia Dictionary Quotes Categories ... Conjectures Riemann hypothesis RH directs here. RH is also the common abbreviation for the soap opera Ryan's Hope In mathematics , the Riemann hypothesis , first formulated by Bernhard Riemann in , is one of the most famous of all unsolved problems. It has been an open question for well over a century, despite attracting concentrated efforts from many outstanding mathematicians. Unlike some other celebrated problems, it is more attractive to professionals in the field than to amateurs. The Riemann hypothesis is a conjecture about the distribution of the zeros of the Riemann zeta function s ). The Riemann zeta function is defined for all complex numbers s s s s The real part of any non-trivial zero of the Riemann zeta function is 1/2. Thus the non-trivial zeros should lie on the so-called critical line it with t a real number and i the imaginary unit . The Riemann zeta function along the critical line is sometimes studied in terms of the Z function , whose real zeros correspond to the zeros of the zeta function on the critical line.

19. Wolfram|Alpha: Riemann Hypothesis : Statement, Proof, Status, Awards, ... Complete information and computations for Riemann hypothesis basic properties, history, http://www.wolframalpha.com/entities/famous_math_problems/riemann_hypothesis/t8/

document.write(''); Home Examples About FAQS ... Media Resources A Wolfram Web Resource Wolfram Research Wolfram Mathematica Wolfram Demonstrations ... Wolfram Tones Enter something to compute or figure out Calculate Riemann hypothesis: statement, proof, status, awards, formulation, ... Input interpretation: Statement: Documentation Properties Definition Documentation ... Definition Formal statement: Documentation Definition Documentation Definition Alternate names: History: More Current evidence: Associated prizes: Computed by Wolfram Mathematica Download as: PDF Live Mathematica A Wolfram Research Company Participate Entity Index Riemann hypothesis: statement, proof, status, awards, formulation, ... Input: riemann hypothesis Riemann hypothesis Statement The nontrivial zeros of the Riemann zeta function zeta(s) all lie on the critical line Re(s) = 1/ Formal statement Alternate names Hilbert(close curly quote)s eighth problem RH Smale(close curly quote)s first problem History Current evidence It has been verified that the first nontrivial zeros of the zeta function lie on the critical line. Associated prizes

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