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Topic: Riemann hypothesis

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Riemann zeta function

Generalized Riemann hypothesis

On the Number of Primes Less Than a Given Magnitude

Riesz function

Riemann integral

Prime number theorem

List of conjectures

Riemann hypothesis for curves over finite fields

Riemann surface

Bernhard Riemann

Landau symbol

Riemann

Prime number

Number theory

Surreal number

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	Riemann hypothesis - Wikipedia, the free encyclopedia
		Riemann mentioned the conjecture that became known as the Riemann hypothesis in his 1859 paper On the Number of Primes Less Than a Given Magnitude, but as it was not essential to his central purpose in that paper, he did not attempt a proof.
		The zeroes of the Riemann zeta-function and the prime numbers satisfy a certain duality property, known as the explicit formulae, which shows that in the language of Fourier analysis the zeros of the Riemann zeta-function can be regarded as the harmonic frequencies in the distribution of primes.
		The Riemann hypothesis is equivalent to certain conjectures of group theory.
	en.wikipedia.org /wiki/Riemann_Hypothesis (2351 words)

	Encyclopedia :: encyclopedia : Riemann hypothesis (Site not responding. Last check: 2007-10-22)
		In mathematics, the Riemann hypothesis (also called the Riemann zeta hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous of all unsolved problems.
		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 zeroes of the Riemann zeta function and the prime numbers satisfy a certain duality property, known as the explicit formulae which show that in the language of Fourier analysis the zeros of the zeta function can be regarded as the harmonic frequencies in the distribution of primes.
	www.hallencyclopedia.com /Riemann_hypothesis (1834 words)

	Generalized Riemann hypothesis - Wikipedia, the free encyclopedia
		The Riemann hypothesis is one of the most important conjectures in mathematics.
		When the Riemann hypothesis is formulated for Dedekind zeta-functions, it is known as the extended Riemann hypothesis and when it is formulated for Dirichlet L-functions, it is known as the generalized Riemann hypothesis.
		The generalized Riemann hypothesis (for Dirichlet L-functions) was probably formulated for the first time by Piltz in 1884.
	en.wikipedia.org /wiki/Generalized_Riemann_hypothesis (776 words)

	Riemann hypothesis
		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.
		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 is proved true, it could open an entirely new window on the nature of reality and the relationship between the abstract world of mathematics and the behavior of matter and energy.
	www.daviddarling.info /encyclopedia/R/Riemann_hypothesis.html (462 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		The Riemann hypothesis, first formulated by Bernhard Riemann in 1859, is a conjecture about the distribution of the zeros of Riemann's zeta function ζ(s).
		The zeros of the Riemann zeta function and the prime numbers satisfy a certain duality property, known as the explicit formulae which show that in the language of Fourier analysis the zeros of the zeta function can be regarded as the harmonic frequencies in the distribution of primes.
		Subsequent work has strongly born out this discovery, and the distribution of the zeros of the Riemann zeta function is now believed to satisfy the same statistics as the eigenvalues of a random Hermitian matrix, the statistics of the so-called Gaussian Unitary Ensemble.
	wikiwhat.com /encyclopedia/r/ri/riemann_hypothesis.html (586 words)

	Online Encyclopedia and Dictionary - Riemann hypothesis
		In mathematics, the Riemann hypothesis, first formulated by Bernhard Riemann in 1859, is one of the most famous of all unsolved problems.
		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.
		Riemann knew that the non-trivial zeros of the zeta function were symmetrically distributed about the line z = 1/2 + it, and he knew that all of its non-trivial zeros must lie in the range 0 ≤ Re(z) ≤ 1.
	www.fact-archive.com /encyclopedia/Riemann_hypothesis (1389 words)

	BERNHARD RIEMANN
		Riemann was born in 1826 in the kingdom of Hannover, later part of Germany.
		Riemann's essay made considerable progress on this problem, first by giving a criterion for a function to be integrable (or as we now say, Riemann integrable), and then by obtaining a necessary condition for a Riemann integrable function to be representable by a Fourier series.
		Riemann's lecture, "On the hypotheses that lie at the foundation of geometry" was given on June 10, 1854.
	www.usna.edu /Users/math/meh/riemann.html (1057 words)

	Riemann
		The basic mathematical structure of quantum TGD led a couple of years ago to a sharpening of the Riemann hypothesis stating that the zeros of zeta are of form x=1/2+iy and p^{iy} is a rational phase for every prime and thus defines Pythagorean triangle (orthogonal triangle with integer-valued sides).
		The vanishing of Riemann Zeta reduces to an orthogonality condition for the eigenfunctions of a non-Hermitian operator D^+ having the zeros of Riemann Zeta as its eigenvalues.
		Riemann hypothesis follows by reductio ad absurdum from the hypothesis that ordinary superconformal algebra acts as gauge symmetries for all coherent states orthogonal to the vacuum state, including also the non-physical might-be coherent states off from the critical line.
	www.physics.helsinki.fi /~matpitka/Riema.html (1332 words)

	Biography of Riemann
		Riemann submitted his thesis in 1851 to Gauss, an impressed Gauss said that Riemann possessed a "Gloriously fertile originality." Thanks to Gauss's recommendation Riemann was appointed to a position at Göttingen.
		The Riemann Hypothesis states that the nontrivial roots of the Riemann zeta function (which is explained later in the web page) defined on the complex plane C all have real part 1/2.
		Riemann observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function.
	www.andrews.edu /~calkins/math/biograph/bioriema.htm (912 words)

	PlanetMath: Riemann zeta function
		The Riemann zeta function is defined to be the complex valued function given by the series
		If true, the hypothesis would have profound consequences on the distribution of primes in the integers [5].
		This is version 12 of Riemann zeta function, born on 2002-05-06, modified 2005-03-15.
	planetmath.org /encyclopedia/RiemannHypothesis.html (722 words)

	Open Questions: The Riemann Hypothesis
		The Riemann hypothesis is the statement that the zeros of a certain complex-valued function ζ(s) of a complex number s all have a certain special form.
		Riemann (1826-66) studied the zeta function (including Euler, as we shall see), the notation is Riemann's, and hence the function is commonly known as the zeta function, after the greek letter ζ.
		Riemann's contribution, however, was to realize that ζ(s) could be analytically continued to the whole complex plane, to derive many important properties of ζ(s), such as its functional equation, and -- most importantly -- to suspect its deeper relationship to the distribution of prime numbers.
	www.openquestions.com /oq-ma014.htm (14106 words)

	The Riemann Hypothesis
		Put simply, the Riemann Hypothesis is a prediction about the distribution of zeros in Riemann’s zeta function.
		Riemann had discovered a crucial link between the zeros in his zeta function and the prime numbers.
		If the Riemann hypothesis turns out to be true the implications are huge for mathematics and its applications.
	people.bath.ac.uk /ejwt20/RiemannHypothesis.htm (395 words)

	Prime Time - Mathematicians have tried in vain to this day to discover some oreder inthe sequence of prime numbers...
		Riemann decided to see what would happen if he fed the zeta function complex numbers—numbers made from a real part (an ordinary number) and a so-called imaginary part (a multiple of i, the square root of -1).
		Riemann worked out that if the zeros really do lie on the critical line, then the primes stray from the 1/ln(x) distribution exactly as much as a bunch of coin tosses stray from the 50:50 distribution law.
		To prove the Riemann hypothesis, researchers must pinpoint a specific quantum system whose energy levels correspond exactly to the zeros, and prove that they do so all the way to infinity.
	www.timetoeternity.com /time_space_light/prime_time.htm (2658 words)

	Jascha Hoffman: Prime Time
		The Riemann Hypothesis concerns the prime numbers, which have been recognized as the “atoms of arithmetic” since ancient times but have remained much more elusive than that metaphor would imply.
		The Riemann Hypothesis is, roughly speaking, a 150-year-old guess about how the primes are spaced along the number line.
		While a solution to the Riemann Hypothesis would not mean the end of public-key encryption, there is a real threat that related advances in our understanding of the primes could spell catastrophe for electronic commerce and national security.
	www.bostonreview.net /BR29.2/hoffman.html (1980 words)

	Riemann hypothesis - Search Results - MSN Encarta
		Riemann, Georg Friedrich Bernhard (1826-66), German mathematician, who developed a system of geometry that aided the development of modern...
		Hypothesis, a preliminary assumption or tentative explanation that accounts for a set of facts, taken to be true for the purpose of investigation...
		Evolution, in biology, complex process by which the characteristics of living organisms change over many generations as traits are passed from one...
	encarta.msn.com /encnet/refpages/search.aspx?q=Riemann+hypothesis (141 words)

	The Riemann Hypothesis
		The unproved Riemann hypothesis is that all of the nontrivial zeros are actually on the critical line.
		Another way to generalize Euler's sum is to leave the field of rational numbers, and replace the denominators with the norms of the non-zero ideals (special sets of elements) in a finite field extention of the rationals K (called a number field).
		The generalized Riemann Hypothesis is again that the zeros in the critical region all have real part 1/2.
	primes.utm.edu /notes/rh.html (735 words)

	ZetaGrid - Verification of the Riemann Hypothesis
		The verification of Riemann's Hypothesis (formulated in 1859) is considered to be one of modern mathematic's most important problems.
		The Riemann Hypothesis asserts that all nontrivial zeros of the zeta function are on the critical line (1/2+it where t is a real number).
		The result of the computation which verified the first 100 billion zeros of the Riemann zeta function confirms the calculations made previously by other scientists and extends these to the first 100 billion nontrivial zeros.
	www.zetagrid.net /zeta/rh.html (315 words)

	riemann (Site not responding. Last check: 2007-10-22)
		The Riemann Hypothesis is currently the most famous unsolved problem in mathematics.
		See the Riemann Zeta Function in the CRC Concise Encyclopedia of Mathematics for more information on this.
		The Riemann Hypothesis : all nontrivial roots of the Zeta function are of the form (1/2 + b I).
	www.mathpuzzle.com /riemann.html (309 words)

	The Prime Glossary: Riemann hypothesis (Site not responding. Last check: 2007-10-22)
		Riemann noted that his zeta function had trivial zeros at -2, -4, -6,...
		In fact the classical proofs of the prime number theorem require an understanding of the zero free regions of this function, and in 1901 von Koch showed that the Riemann hypothesis is equivalent to:
		Because of this relationship to the prime number theorem, Riemann's hypothesis is easily one of the most important conjectures in prime number theory.
	primes.utm.edu /glossary/page.php?sort=RiemannHypothesis (111 words)

	Riemann Hypothesis in a Nutshell (Site not responding. Last check: 2007-10-22)
		The Riemann Hypothesis (RH) is that all non-trivial zeros of the zeta function lie on the critical line.
		The function Z(t) is typically the object of study for locating zeros of the zeta function on the critical line and verifying the Riemann Hypothesis.
		This alternation of zeros with Gram points is key to verifying the Riemann Hypothesis.
	web.mala.bc.ca /pughg/RiemannZeta/RiemannZetaLong.html (1384 words)

	The Riemann Hypothesis
		This covers all the number theory necessary for a basic understanding of the Riemann Hypothesis, which is covered in its final section.
		Riemann's Zeros: The Search for the $1 Million Solution to the Greatest Problem in Mathematics (Atlantic Books, 2002) - a recently published popular account of the Riemann hypothesis, to be published in the U.S. in April as The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics (Farrar, Straus and Giroux)
		Professor J.E. Littlewood's brief argument as to why he believes the Riemann Hypothesis to be false.
	www.maths.ex.ac.uk /~mwatkins/zeta/riemannhyp.htm (1000 words)

	THE RIEMANN HYPOTHESIS (Site not responding. Last check: 2007-10-22)
		The remaining part of Riemann’s paper is very obscure and confusing because of its excessive brevity.
		The hypothesis 2), with error term 0(log T) was proved in 1894 by von Mangoldt, who also proved hypothesis 6) (but he used an alternative way).
		This series converges very rapidly and one might suppose that an approximation to the truth could be obtained by replacing it by their first terms.
	www.cuatrovientos.com.ar /_vti_z (513 words)

	Riemann Hypothesis
		The famous Riemann hypothesis concerns the locations of the remaining, ‘nontrivial’ zeros of
		The conjecture that all these zeros are in fact on the critical line is the Riemann hypothesis.
		Riemann briefly mentions his attempts to prove all the zeros were on the critical line, but resigns to put this task aside for another time.
	www.physicsforums.com /showthread.php?t=78799 (2378 words)

	The Riemann Hypothesis (Site not responding. Last check: 2007-10-22)
		The Riemann zeta function is of central importance in the study of prime numbers.
		The case Re(z)=1 was proved in 1896 by Hadamard and de la Vallée-Poussin and used in their proof of the Prime Number Theorem.
		Riemann conjectured that all nontrivial zeros are at Re(z)=1/2.
	users.forthnet.gr /ath/kimon/Riemann/Riemann.htm (289 words)

	Proof For Riemann Hypothesis?
		The spirited competition to prove the hypothesis - which carries a $1 million prize for whomever accomplishes it first - has encouraged de Branges to announce his work as soon as it was completed rather than go through the more traditional peer reviewed publishing process.
		The Riemann hypothesis is a highly complex theory about the nature of prime numbers - those numbers divisible only by 1 and themselves - that has stymied mathematicians since 1859.
		A mathematician from Purdue University claims to have proven the Riemann hypothesis, often dubbed the greatest unsolved problem in mathematics.
	www.scienceagogo.com /news/20040508233713data_trunc_sys.shtml (878 words)

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