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	Riemann hypothesis - Wikipedia, the free encyclopedia
		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.
		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 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.wikipedia.org /wiki/Riemann_hypothesis (1839 words)

	Bernhard Riemann - Wikipedia, the free encyclopedia
		Georg Friedrich Bernhard Riemann (September 17, 1826 - July 20, 1866) (pronounced REE mahn) was a German mathematician who made important contributions to analysis and differential geometry, some of them paving the way for the later development of general relativity.
		His name is connected with the Riemann zeta function, the Riemann integral, the Riemann lemma, Riemannian manifolds, the Riemann mapping theorem, Riemann-Hilbert problems, Riemann surfaces, the Riemann-Roch theorem, the Riemann sphere, and the Cauchy-Riemann equations.
		Riemann held his first lectures in 1854, which not only founded the field of Riemannian geometry but set the stage for Einstein's general relativity.
	en.wikipedia.org /wiki/Bernhard_Riemann (346 words)

	Riemann
		In the spring of 1846 Riemann enrolled at the University of Göttingen.
		Riemann moved from Göttingen to Berlin University in the spring of 1847 to study under Steiner, Jacobi, Dirichlet and Eisenstein.
		In 1859 Dirichlet died and Riemann was appointed to the chair of mathematics at Göttingen on 30 July.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Riemann.html (2606 words)

	Johanneum Lüneburg Bernhard Riemann
		Bernhard Riemann was born on the 17th September 1826 in Breselenz/Dannenberg where his father was a vicar.
		The teachers' recommendation that Riemann was "because of his abilities definitely suitable for the study of mathematical sciences" was not appreciated by his father.
		I hope to have given you an idea of the person Bernhard Riemann, and to have imparted to you a presentiment of the depth of his mathematical thought, although his importance as a mathematician is mainly based upon abstract foundations, with no respect to intellegibility.
	rzserv2.fh-lueneburg.de /u1/gym03/englpage/chronik/riemann/riemann.htm (1706 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
		However, Riemann's thesis is a strikingly original piece of work which examined geometric properties of analytic functions, conformal mappings and the connectivity of surfaces.
		In the mathematical apparatus developed from Riemann's address, Einstein found the frame to fit his physical ideas, his cosmology, and cosmogony: and the spirit of Riemann's address was just what physics needed: the metric structure determined by data.
		A newly elected member of the Berlin Academy of Sciences had to report on their most recent research and Riemann sent a report on On the number of primes less than a given magnitude another of his great masterpieces which were to change the direction of mathematical research in a most significant way.
	www.meta-religion.com /Mathematics/Biography/riemann.htm (2617 words)

	Riemann hypothesis at opensource encyclopedia (Site not responding. Last check: 2007-11-06)
		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).
		Riemann knew that the non-trival 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.
		Subsequent work has strongly borne 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.
	wiki.tatet.com /Riemann_hypothesis.html (904 words)

	AllRefer.com - Bernhard Riemann (Mathematics, Biography) - Encyclopedia
		He laid the foundations of a non-Euclidean system of geometry (Riemannian geometry) representing elliptic space and generalized to n dimensions the work of C. Gauss in differential geometry, thus creating the basic tools for the mathematical expression of the general theory of relativity.
		Riemann also was interested in mathematical physics, particularly optics and electromagnetic theory.
		The so called "Riemann hypothesis," concerning the instances in which the function's value is zero, is one of the great unsolved problems in mathematics.
	reference.allrefer.com /encyclopedia/R/Riemann.html (282 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
		Georg Friedrich Bernhard Riemann was born on September 17, 1826 in Breselenz, Germany to Georg Friedrich Bernhard Riemann and Charlotte Ebell.
		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.
		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)

	Ipotesi di Riemann: Tutte le informazioni su Ipotesi di Riemann su Encyclopedia.it (Site not responding. Last check: 2007-11-06)
		L'ipotesi o congettura di Riemann, fu formulata la prima volta dal matematico di Gottinga Bernhard Riemann nel 1859.
		Molti matematici pensano che l'ipotesi di Riemann sia vera (non tutti però, J.
		Pare che Riemann avesse risolto la congettura che porta il suo nome, ma purtroppo le sue carte furono distrutte; non possiamo quindi sapere per certo se egli avesse solo impostato o risolto quel mistero.
	www.encyclopedia.it /i/ip/ipotesi_di_riemann.html (409 words)

	Johanneum Lüneburg Bernhard Riemann
		Manche Biographen sehen in der Unterernährung in der Jugendzeit einen Grund für den frühen Tod Riemanns, seiner Elten und mehrerer seiner Geschwister.
		Riemanns Habilitationsvortrag 1854 enthielt Erkenntnisse, die ihm einen bleibenden Platz nicht nur unter den Mathematikern, sondern auch unter den Wegbereitern der wissenschaftlichen Weltanschauung sicherte.
		Ich hoffe, daß ich Ihnen den Menschen Bernhard Riemann nahe gebracht habe, daß ich Ihnen eine Ahnung von der Tiefe seiner mathematischen Gedanken vermitteln konnte, obwohl seine Bedeutung als Mathematiker gerade darin lag, daß er ohne Rücksicht auf Anschaulichkeit abstrakte Grundlagen legte.
	www.fh-lueneburg.de /u1/gym03/homepage/chronik/riemann/riemann.htm (1515 words)

	Bernhard Riemann (Site not responding. Last check: 2007-11-06)
		Georg Friedrich Bernhard Riemann (September 17, 1826 - June 20, 1866) was a German mathematician who made important contributions to analysis and differential geometry, some of them paving the way for the later development of general relativity.
		His name is connected with the zeta function, the Riemann integral, the Riemann lemma, Riemannian manifolds and Riemann surfaces.
		He was promoted an extraordinary professor at the University of Göttingen in 1857 and became an ordinary professor in 1859.
	www.wikiverse.org /bernhard-riemann (296 words)

	Riemann Package (Site not responding. Last check: 2007-11-06)
		Riemann is a Maple package which allows the user to perform indicial manipulation of user-defined or built-in tensors.
		Riemann's ideas were used by A.Einstein as the mathematical background of the General Relativity theory.
		The Riemann package is included in the share library of Maple V release 5.
	www.astro.queensu.ca /~portugal/Riemann.html (432 words)

	The Riemann Hypothesis (Site not responding. Last check: 2007-11-06)
		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)

	Nat' Academies Press, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003)
		Riemann tackled the problem with the most sophisticated mathematics of his time, using tools that even today are taught only in advanced college courses, and inventing for his purposes a mathematical object of great power and subtlety.
		Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics One would, of course, like to have a rigorous proof of this, but I have put aside the search for such a proof after some fleeting vain attempts because it is not necessary for the immediate objective of my investigation.
		The Riemann Hypothesis, as that guess came to be called, remained an obsession all through the twentieth century and remains one today, having resisted every attempt at proof or disproof.
	www.nap.edu /books/0309085497/html/R1.html (2237 words)

	ZetaGrid - Verification of the Riemann Hypothesis (Site not responding. Last check: 2007-11-06)
		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)

	Amazon.com: Books: The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics (Site not responding. Last check: 2007-11-06)
		Riemann's hypothesis is not easily grasped; what Sabbagh wants to do is to enhance your understanding of it.
		The hypothesis itself is an outcome of Riemann's zeta function which is the sum of the series 1 + 1/2^s +1/3^s...1/n^s, which means 1 + 1/2^a+ib + 1/3^a+ib (where i is an imaginary number).
		These zeroes, as its turns out, fall on what is known as the "critical strip" and their graph is linked to the fluctation of the primes, which are themselves the building blocks for all the other numbers.
	www.amazon.com /exec/obidos/tg/detail/-/0374250073?v=glance (2531 words)

	7.1. Riemann Integral (Site not responding. Last check: 2007-11-06)
		Riemann sums have the practical disadvantage that we do not know which point to take inside each subinterval.
		The third example shows that not every function is Riemann integrable, and the second one shows that we need an easier condition to determine integrability of a given function.
		Suppose f is Riemann integrable over an interval [-a, a] and f is an odd function, i.e.
	web01.shu.edu /projects/reals/integ/riemann.html (1802 words)

	Hugo Riemann
		Weil Leipzig seine "Über das musikalische Hören" nicht annehmen wollte Riemann damit in Göttingen.
		Dann habilitierte er sich 1878 doch noch an der Universität in mit den "Studien zur Geschichte der Notenschrift".
		Im Jahre 1880 übernahm Riemann als Dirigent den gemischten in Bromberg und war gleichzeitig Privatdozent in Leipzig Des weiteren wirkte er als Theorielehrer an Konservatorien in Hamburg (1880-1891) Sondershausen (1890) und Wiesbaden (1890-1895).
	www.uni-protokolle.de /Lexikon/Hugo_Riemann.html (381 words)

	riemann (Site not responding. Last check: 2007-11-06)
		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)

	Read This: Bernhard Riemann, 1826-1866
		Bernhard Riemann (1826-1866) is a central figure in the history of mathematics.
		This theme, captured in the book's subtitle and repeated as the title of Chapter 4, is that Riemann is the person most responsible for turning mathematics from a principally algorithmic science to a principally conceptual science.
		Riemann's 1851 thesis is on the foundations of one-variable complex analysis, through what we now call the Riemann mapping theorem.
	www.maa.org /reviews/riemann.html (1284 words)

	Riemann, Bernhard -- Encyclopædia Britannica
		in full Georg Friedrich Bernhard Riemann German mathematician whose profound and novel approaches to the study of geometry laid the mathematical foundation for Albert Einstein's theory of relativity.
		It is easy to define the area of a shape whose edges are straight: for example, the area of a rectangle is just the product of the lengths of two...
		The real significance of Lobachevsky's geometry was not fully understood and appreciated until the work of the great German mathematician Bernhard Riemann on the foundations of geometry (1868) and the proof of the consistency of non-Euclidean geometry by his compatriot Felix Klein in 1871.
	www.britannica.com /eb/article-9063646 (818 words)

	Bernhard Riemann
		Sehr früh war Riemann von den Primzahlen und ihren Eigenschaften fasziniert.
		Diese berühmte Vermutung findet sich in seinem Werk 'Über die Anzahl der Primzahlen unter einer gegebenen Grösse' das Riemann 1849 als Dreiundzwanzigjähriger veröffentlichte.
		Riemann zeigte auf, dass es in gleicher Weise, wie es verschiedene Arten von Kurven und Flächen gibt, auch verschiedene Arten von dreidimensionalen Räumen gibt.
	www.mathematik.ch /mathematiker/riemann.php (216 words)

	The Riemann Hypothesis (Site not responding. Last check: 2007-11-06)
		It is now unquestionably the most celebrated problem in mathematics and it continues to attract the attention of the best mathematicians, not only because it has gone unsolved for so long but also because it appears tantalizingly vulnerable and because its solution would probably bring to light new techniques of far reaching importance.
		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)
	www.maths.ex.ac.uk /~mwatkins/zeta/riemannhyp.htm (1000 words)

	The Riemann Hypothesis
		Riemann noted that his zeta function had trivial zeros at -2, -4, -6,...
		The unproved Riemann hypothesis is that all of the nontrivial zeros are actually on the critical line.
		The error term depended on what was known about the zero-free region of the Riemann zeta function within the critical strip.
	primes.utm.edu /notes/rh.html (452 words)

	Riemann Hypothesis in a Nutshell (Site not responding. Last check: 2007-11-06)
		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.
	www.math.ubc.ca /~pugh/RiemannZeta/RiemannZetaLong.html (1384 words)

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