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Topic: Generalized Riemann hypothesis

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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 Riemann hypothesis is equivalent to certain conjectures of group theory.
	en.wikipedia.org /wiki/Riemann_hypothesis (1839 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 was probably formulated for the first time by Piltz in 1884.
	en.wikipedia.org /wiki/Generalized_Riemann_hypothesis (698 words)

	Riemann hypothesis - Open Encyclopedia (Site not responding. Last check: 2007-10-21)
		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.
		The practical uses of the Riemann hypothesis include many equations that have been 'solved' in abstract mathematics with the assumption of the Riemann hypothesis.
	open-encyclopedia.com /RH (1002 words)

	Riemann hypothesis -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		The Riemann hypothesis is a (Reasoning that involves the formation of conclusions from incomplete evidence) conjecture about the distribution of the (The quantity that registers a reading of zero on a scale) zeros of the (Click link for more info and facts about Riemann zeta function) Riemann zeta function ζ(s).
		The Riemann zeta function is defined for all (A number of the form a+bi where a and b are real numbers and i is the square root of -1) complex numbers s ≠ 1.
		The Riemann zeta function along the critical line is sometimes studied in terms of the (Click link for more info and facts about Z function) Z function, whose real zeros correspond to the zeros of the zeta function on the critical line.
	www.absoluteastronomy.com /encyclopedia/R/Ri/Riemann_hypothesis.htm (1577 words)

	Riemann hypothesis: Definition and Links by Encyclopedian.com - All about Riemann hypothesis (Site not responding. Last check: 2007-10-21)
		However Selberg in the early 1950s proved a duality between the length spectrum of a Riemann surface and the eigenvalues of its Laplacian.
		Dyson saw that the statistical distribution found by Montgomery was exactly the same as the pair correlation distribution for the eigenvalues of a random Hermitian matrix.
		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[?].
	www.encyclopedian.com /ri/Riemann-hypothesis.html (598 words)

	Riemann hypothesis Article, Riemannhypothesis Information (Site not responding. Last check: 2007-10-21)
		The Riemann hypothesis, first formulated by Bernhard Riemann in 1859, is a conjecture about the distribution of the zeros of Riemann's zeta functionζ(s).
		Riemann mentioned the conjecture that became known as the Riemann hypothesis in his 1859 paper On theNumber of Primes Less Than a Given Magnitude, but as it was not essential to his central purpose in that paper, he didnot attempt a proof.
		The zeros of the Riemann zeta function and the prime numbers satisfy a certain duality property, known as the explicitformulae 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.anoca.org /zeros/proof/riemann_hypothesis.html (891 words)

	Riemann hypothesis - Encyclopedia, History, Geography and Biography
		In mathematics, the Riemann hypothesis (aka Riemann zeta hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous of all unsolved problems.
		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.
		Another conjecture is the large prime gap conjecture; Cramér proved that on the Riemann hypothesis we have that the largest gaps between successive prime numbers is $O(\sqrt{p} \ln p)$ .
	www.arikah.com /encyclopedia/Riemann_hypothesis (1479 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)

	Riemann hypothesis (Site not responding. Last check: 2007-10-21)
		The Riemann hypothesis, first formulated by Bernhard Riemann in 1859, is a conjecture about the distribution of the zeross 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
		Recent work has focussed on the explicit calculation of the locations of large numbers of zeros (in the hope of finding a counterexample) and placing upper bounds on the proportion of zeros that can lie away from the critical line (in the hope of reducing this to zero).
	www.sciencedaily.com /encyclopedia/riemann_hypothesis (934 words)

	[No title]
		General informations about these constants can be found on Eric Weissten's world of mathematics site.
		Riemann himself gave several methods ; later, mathematicians like Hardy, Siegel and others, enriched the list of proofs (see for example [4], where seven different methods are presented).
		Interestingly, disproof of the Riemann hypothesis (e.g., by using a computer to actually find a zero off the critical line), does not earn the one million dollars award.
	numbers.computation.free.fr /Constants/Miscellaneous/zetageneralities.html (1521 words)

	Miller-Rabin primality test - Wikipedia, the free encyclopedia
		The Miller-Rabin primality test is a primality test: an algorithm which determines whether a given number is prime, similar to the Fermat primality test and the Solovay-Strassen primality test.
		Its original version, due to G. Miller, is deterministic, but it relies on the unproven generalized Riemann hypothesis; M.
		Especially in cryptographic application an adversary might try to send you a pseudoprime in a place where a prime number is required.
	www.wikipedia.org /wiki/Miller-Rabin_primality_test (751 words)

	Ideal Class Groups
		Using these relations, a generating set for the ideal class group is derived (via matrix echelonization), and in the final step it is verified that the correct orders for the generators have been found.
		An integral upper bound for norms of generators of the ideal class group for K or O assuming the generalized Riemann hypothesis.
		Let a_i be the generators for the cyclic factors of the class group of O. This function returns generators for a_i^(c_i) where c_i is the order of a_i in the class group.
	www.math.niu.edu /help/math/magmahelp/text667.html (921 words)

	Riemann hypothesis
		In June 2004, Louis De Branges de Bourcia claimed to have proved the Riemann hypothesis but this has not yet been confirmed (see below).
		The full purported proof is "Riemann Zeta functions" [1](pdf).
		ZetaGrid A distributed computing project that has verified the Riemann Hypothesis for the first 793 billion nontrivial zeros.
	www.brainyencyclopedia.com /encyclopedia/r/ri/riemann_hypothesis.html (916 words)

	Grand Riemann hypothesis - Wikipedia, the free encyclopedia
		In mathematics, the grand Riemann hypothesis is a generalisation of the Riemann hypothesis and Generalized Riemann hypothesis.
		It states that the nontrivial zeros of all automorphic L-functions lie on the critical line 1/2 + it with t a real number and i the imaginary unit.
		The modified grand Riemann hypothesis is the assertion that the nontrivial zeros of all automorphic L-functions lie on the critical line or the real line.
	en.wikipedia.org /wiki/Grand_Riemann_hypothesis (136 words)

	17a (Site not responding. Last check: 2007-10-21)
		The Generalized Riemann Hypothesis(GRH) is the assertion that the Riemann Hypothesis is true, and in addition the nontrivial zeros of all Dirichlet $L$-functions lie on the critical line
		Equivalently, GRH asserts that the nontrivial zeros of all degree 1
		The Modified Generalized Riemann Hypothesis(MGRH) is the assertion that the Riemann Hypothesis is true, and in addition the nontrivial zeros of all Dirichlet $L$-functions lie either on the critical line
	www.aimath.org /WWN/rh/articles/html/17a (78 words)

	[No title]
		The generalized Riemann Hypothesis represents an extension of the Riemann hypothesis, which applies to the Riemann zeta function, to the class of functions known as the Dirichlet L-functions.
		Along with the Birch-Swinnerton-Dyer conjecture, the generalized Riemann Hypothesis is in fact the basis of this research.
		In attempting to test the generalized Riemann Hypothesis via a “different” approach, two functions Ns and Ss, which computed the normalized and “smooth” sums of ap ‘s, were created.
	www.math.princeton.edu /mathlab/projects/ellcurves/op/curvecubic.doc (2824 words)

	PlanetMath: generalized Riemann hypothesis
		This generalization of the Riemann hypothesis to arbitrary Dedekind zeta functions states that for any number field
		Cross-references: number field, Dedekind zeta functions, Riemann hypothesis
		This is version 2 of generalized Riemann hypothesis, born on 2003-08-28, modified 2003-08-29.
	planetmath.org /encyclopedia/GeneralizedRiemannHypothesis.html (57 words)

	ICM 94: Abstract (Site not responding. Last check: 2007-10-21)
		A very much stronger bound, known as the Lindelöf Hypothesis, is expected to hold, and indeed is known to follow as a simple consequence of the Generalized Riemann Hypothesis.
		This is the main goal of the study but, for a number of applications, it is not really necessary to have such a strong bound but is crucial to have one which improves the convexity estimate.
		These may be generalized to a great extent to other such functions in studying the "$s$-aspect" but not to the (usually more important) problem of studying the other parameters in which these more general functions vary.
	www.ams.org /mathweb/icm94/03.friedlander.html (400 words)

	[No title]
		This conjecture, known as the Riemann hypothesis, has never been proved or disproved, and is probably the most important unsolved problem in mathematics.
		The Riemann hypothesis makes the zeta function so famous, and numerical computation have been made to check it for various sets of zeros.
		Andrew Odlyzko papers on the Riemann Zeta Function and related topics.
	numbers.computation.free.fr /Constants/Miscellaneous/zeta.html (229 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		I have heard it > implies many things, including a polynomial time nonprobabilistic > primality testing algorithm and the existence of small quadratic > nonresidues modulo primes(under 2(ln p)^2 where p is the prime).
		Like the zeta function it has an Euler product expansion prod_p (1 - chi(p)p^{-s})^{-1}, analytic continuation to the whole complex plane with a functional equation and so on (although if chi is non-trivial it does not have a pole at s=1).
		The GRH, apparently first written down by Piltz, 1884, is that every such function has its zeroes in the criticial strip 0 <= re.s <= 1 on the line re.s = 1/2.
	www.math.niu.edu /~rusin/papers/known-math/97/grh (181 words)

	The Riemann Hypothesis (Site not responding. Last check: 2007-10-21)
		An observation on the zero-free region of the Riemann zeta-function
		The Riemann hypothesis for polynomials orthogonal on the unit circle.
		On a necessary condition for the validity of the Riemann hypothesis for functions that generalize the Riemann zeta function.
	mathews.ecs.fullerton.edu /c2003/RiemannHypothesisBib/Links/RiemannHypothesisBib_lnk_3.html (889 words)

	[No title]
		It is well-known that if the class number of some imaginary quadratic field with large discriminant is one then we will have an egregious counterexample to the Generalized Riemann Hypothesis (that is, a zero of the associated Dirichlet L-function which is very close to 1, a weak consequence of the Generalized Riemann Hypothesis).
		A-2 (though not all) then perhaps still the Generalized Riemann Hypothesis is false, though perhaps not with a zero quite so close to 1.
		This is a generalization of the notion of RD types that have been so completely studied (see chapter 3 of my book Quadratics cited below for a complete background on RD types).
	www.math.ucalgary.ca /~ramollin/research2.html (1412 words)

	Riemann Hypothesis (Site not responding. Last check: 2007-10-21)
		The Riemann Hypothesis: The Most Important Unsolved Problem in Mathematics...
		APOLOGY FOR THE PROOF OF THE RIEMANN HYPOTHESIS...
		Crooked Timber » » Riemann hypothesis proved ?...
	www.scienceoxygen.com /math/750.html (121 words)

	Nat' Academies Press, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003)
		In the space available to me, and by the methods of exposition I have decided on, it has proved necessary to omit entire large regions of inquiry relevant to the Riemann Hypothesis.
		You will find not one word here about the Density Hypothesis, the approximate functional equation, or the whole fascinating issue—just recently come to life after long dormancy—of the moments of the zeta function.
		Nor is there any mention of the Generalized Riemann Hypothesis, the Modified Generalized Riemann Hypothesis, the Extended Riemann Hypothesis, the Grand Riemann Hypothesis, the Modified Grand Riemann Hypothesis, or the Quasi-Riemann Hypothesis.
	www.nap.edu /books/0309085497/html/R14.html (603 words)

	The Prime Glossary: odd Goldbach conjecture
		In 1923 Hardy and Littlewood [HL23] showed that it follows from the Riemann Hypothesis for all sufficiently large integers.
		In 1937 Vinogradov [Vinogradov37] removed the dependence on the Riemann Hypothesis, and proved that this it true for all sufficiently large odd integers n (but did not quantify "sufficiently large").
		So then, once the Generalized Riemann Hypothesis is proved, the odd Goldbach conjecture will be too.
	primes.utm.edu /glossary/page.php?sort=OddGoldbachConjecture (338 words)

	Riemann hypothesis - Internet-Encyclopedia.com (Site not responding. Last check: 2007-10-21)
		Find riemann hypothesis and more at Lycos Search.
		Read about riemann hypothesis in the free online encyclopedia and dictionary.
		Find riemann hypothesis at one of the best sites the Internet has to offer!
	www.internet-encyclopedia.com /ie/r/ri/riemann_hypothesis.html (1027 words)

	Under the Assumption of the Generalized Riemann Hypothesis Verifying the Class Number Belongs to NP intersect Co-NP ... (Site not responding. Last check: 2007-10-21)
		Under the Assumption of the Generalized Riemann Hypothesis Verifying the Class Number Belongs to NP intersect Co-NP (ResearchIndex)
		Under the Assumption of the Generalized Riemann Hypothesis Verifying the Class Number Belongs to NP intersect Co-NP
		We show that under the assumption of a certain Generalized Riemann Hypothesis the problem of verifying the value of the class number of an arbitrary algebraic number field F of arbitrary degree belongs to the complexity class NP " co \Gamma NP.
	citeseer.ist.psu.edu /67520.html (571 words)

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