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	Riemann zeta function - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-11-03)
		In mathematics, the Riemann zeta function is a function which is of paramount importance in number theory, because of its relation to the distribution of prime numbers.
		Bernhard Riemann realized that the zeta function can be extended by analytic continuation in a unique way to a holomorphic function ζ(s) defined for all complex numbers s with s ≠ 1.
		Although mathematicians regard the Riemann zeta function as being primarily relevant to the "purest" of mathematical disciplines, number theory, it also occurs in applied statistics (see Zipf's law and Zipf-Mandelbrot law), physics, and the mathematical theory of musical tuning.
	encyclopedia.learnthis.info /r/ri/riemann_zeta_function.html (603 words)

	Riemann Zeta Function
		The Riemann zeta function is an extremely important special function of mathematics and physics that arises in definite integration and is intimately related with very deep results surrounding the prime number theorem.
		The derivative of the Riemann zeta function for
		Howson, A. "Addendum to: 'Euler and the Zeta Function' (Amer.
	users.skynet.be /fa956617/math/topics/RiemannZetaFunction.html (1820 words)

	Riemann zeta function - Wikipedia, the free encyclopedia
		In mathematics, the Riemann zeta-function, named after Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers.
		It is this function that is the object of the Riemann hypothesis.
		There are a number of similar relations involving various well-known multiplicative functions; these are given in the article on the Dirichlet series.
	en.wikipedia.org /wiki/Riemann_zeta_function (1519 words)

	PlanetMath: Riemann zeta function
		The Riemann zeta function is defined to be the complex valued function given by the series
		A nontrivial zero of the Riemann zeta function is defined to be a root
		This is version 12 of Riemann zeta function, born on 2002-05-06, modified 2005-03-15.
	planetmath.org /encyclopedia/RiemannZetaFunction.html (783 words)

	a directory of all known zeta functions
		Zeta functions show up in all areas of mathematics and they encode properties of the counted objects which are well hidden and hard to come by otherwise.
		Tamagawa, "On the zeta function of a division algebra", Annals of Mathematics 77 (1963) 387-405.
		"On the poles of topological zeta functions", preprint (2004), 11pp.
	www.maths.ex.ac.uk /~mwatkins/zeta/directoryofzetafunctions.htm (3278 words)

	DMV Summer school on RMT (Site not responding. Last check: 2007-11-03)
		An exciting and relatively recent development in the theory of zeta function is the realization that the imaginary parts of the zeros, when viewed on the scale of their mean separation, seem to have non-trivial statistics which are precisely those appearing in Random Matrix Theory.
		Zeta functions in arithmetic and their spectral statistics by Zeev Rudnick.
		A. Odlyzko The 10^20-th zero of the Riemann zeta function and 175 million of its neighbors, (numerical evidence for several conjectures about the zeros of zeta and the connection to RMT).
	www.math.tau.ac.il /~rudnick/dmv.html (353 words)

	Riemann Hypothesis in a Nutshell
		The functional equation of the zeta function is
		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.
		Since there are infinitely many non-trivial zeros of the zeta function, there is no way you can verify computationally that they all lie on the critical line.
	web.mala.bc.ca /pughg/RiemannZeta/RiemannZetaLong.html (1384 words)

	zeta function (Site not responding. Last check: 2007-11-03)
		The zeta function that has been generalized for complex arguments is used to describe the distribution of primes.
		The function can be generalized as the generalized zeta function.
		The Riemann function can be used to obtain a fairly good approximation of the prime counting function.
	www.2dcurves.com /gamma/gammaz.html (84 words)

	Statistical mechanics: the Riemann zeta function interpreted as a partition function
		In the theory of the distribution of primes, the fundamental object is the Riemann zeta function.
		The probability distributions of the quantum fluctuations of the grand potential and entropy of the gas are computed as a function of temperature and compared, with good agreement, with general predictions obtained from random matrix theory and periodic orbit theory (based on prime numbers).
		The Green's function is defined on a cylinder of radius R and we show that the condition R = a yields the Riemann zeta function as a quantum transition amplitude for the fermion.
	www.maths.ex.ac.uk /~mwatkins/zeta/physics2.htm (7627 words)

	[No title]
		While numerical computations on zeros of the Zeta function have long been focused on RH verification only (to check the RH, isolating the zeros is sufficient so no precise computations of the zeros are needed) it was Odlyzko who the first, computed precisely large consecutive sets of zeros to observe their distribution.
		As referred by Odlyzko in [16] for example, this motivates the GUE hypothesis which is the conjecture that the distribution of the normalized spacing between zeros of the Zeta function is asymptotically equal to the distribution of the GUE eigenvalues.
		It plays an important role in the study of the zeros of the Zeta function, because it was observed that special phenomenon about the zeta function on the critical line occurs when S(t) is large.
	numbers.computation.free.fr /Constants/Miscellaneous/zetazeroscompute.html (2529 words)

	[No title]
		The Riemann hypothesis makes the zeta function so famous, and numerical computation have been made to check it for various sets of zeros.
		Distribution of the zeros of the Riemann Zeta function
		Andrew Odlyzko papers on the Riemann Zeta Function and related topics.
	numbers.computation.free.fr /Constants/Miscellaneous/zeta.html (229 words)

	MP3 of the Riemann Zeta Function at MROB
		Here is a C program I used to compute (approximate) values of the Zeta function and write them to an existing WAV file.
		The actual computation of the zeta function isn't that accurate, because I am using a series that doesn't converge very fast.
		This doesn't converge well enough for locating zeros of the Zeta function but is adequate for creating a sound wave.
	home.earthlink.net /~mrob/pub/ries/zeta.html (450 words)

	Open Questions: The Riemann Hypothesis
		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.
		The numerical values of such functions at special points (and their "residues" at poles) have particular significance in terms of algebraic and arithmetic objects that the functions are associated with.
		It is a consequence of the functional equation of L(s, χ).
	www.openquestions.com /oq-ma014.htm (14106 words)

	Solid Model of Zeta Function Constructed for Studying the Riemann Hypothesis (Site not responding. Last check: 2007-11-03)
		For instance, the zeros of zeta are intimately connected to variations in the distribution of prime numbers.
		Carter decided to make a solid model of the zeta function for several reasons: the zeta function is an important mathematical question that is familiar to most mathematicians, and the zeta function, though rather complicated-looking, is sufficiently easy to form in three dimensions.
		For the zeta function model, Carter and Bailey found that it was extremely difficult to clean the waste material from the narrow valleys.
	www.sdsc.edu /SDSCwire/v1.7/2017.Carter.html (607 words)

	ZETA
		Whittaker's W function is one of the Confluent Hypergeometric func tions.
		The Jacobian function Zeta is related to the Jacobian function The ta.
		The Polygamma function is used for simplification of the ZETA function for some arguments.
	www.zib.de /Symbolik/reduce/help/r38_0450.html (1723 words)

	Ivars Peterson's MathTrek - The Mark of Zeta
		The zeta function can be rewritten as an infinite product (an infinite string of terms multiplied together), where each term in the product involves a different prime number.
		Riemann found that his zeta function is zero for the values -2, -4, -6, and so on.
		Mathematicians are starting to exploit a remarkable link between the Riemann zeta function, prime numbers, and quantum mechanics, specifically the spectrum of energy levels exhibited by a quantum system such as an atom.
	www.maa.org /mathland/mathtrek_6_21_99.html (1349 words)

	TMF at SDSC (Site not responding. Last check: 2007-11-03)
		The Riemann hypothesis is the conjecture that all non-trivial zeros of Riemann's zeta function lie on the "critical line".5+iy.
		The zeta function plays a surprising role in number theory; for instance, the zeros of zeta are intimately connected to variations in the distribution of prime numbers.
		A three-dimensional model of the zeta function has been made on the rapid prototype machine at SDSC.
	www.sdsc.edu /tmf/Examples/Rie/rie.html (164 words)

	Riemann's Zeta Function
		"...a variety of evidence suggests that underlying Riemann's zeta function is some unknown classical, mechanical system whose trajectories are chaotic and without [time-reversal] symmetry, with the property that, when quantised, its allowed energies are the Riemann zeros.
		Rudnick, "Number theoretic background" (This covers all the number theory necessary for a basic understanding of the Riemann Zeta Function, which is covered in its final section.)
		the functional equation of the zeta function and related issues
	www.maths.ex.ac.uk /~mwatkins/zeta/zetafn.htm (475 words)

	Riemann zeta function -- Encyclopædia Britannica
		function useful in number theory for investigating properties of prime numbers.
		More results on "Riemann zeta function" when you join.
		One mathematician who found the presence of Dirichlet a stimulus to research was Riemann, and his few short contributions to mathematics were among the most influential of the century.
	www.britannica.com /eb/article-9063648 (770 words)

	Riemann Zeta Function graphics
		As a complex valued function of a complex variable, the graph of the Riemann zeta function ζ(s) lives in four dimensional real space.
		The real and imaginary parts of ζ(s) are each real valued functions; we can think of the graphs of each one as a surface in three dimensional space.
		The next section shows the converse idea, that is, how the zeros of the Riemann zeta function determine the location of the primes.
	www.math.ucsb.edu /~stopple/zeta.html (931 words)

	Local zeta-function (Site not responding. Last check: 2007-11-03)
		In number theory a local zeta-function is a generating function Z(t) for the number of solutions of set of equations defined over a finite field F in extension fields F
		The analogy with the Riemann zeta function $\zeta(s) comes via consideration of the logarithmic derivative \zeta'(s)/\zeta(s) .$
		In practice it makes Z a rational function of t something that is interesting even in case of V an elliptic curve over finite field.
	www.freeglossary.com /Local_zeta_function (751 words)

	riemann (Site not responding. Last check: 2007-11-03)
		See the Riemann Zeta Function in the CRC Concise Encyclopedia of Mathematics for more information on this.
		A root of a function is a value x such that f(x) = 0.
		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)

	An Introduction to the Theory of the Riemann Zeta-Function - Cambridge University Press (Site not responding. Last check: 2007-11-03)
		This is a modern introduction to the analytic techniques used in the investigation of zeta functions, through the example of the Riemann zeta function.
		Riemann introduced this function in connection with his study of prime numbers and from this has developed the subject of analytic number theory.
		Since then many other classes of ‘zeta function’ have been introduced and they are now some of the most intensively studied objects in number theory.
	www.cambridge.org /uk/catalogue/catalogue.asp?isbn=0521335353 (240 words)

	Math Games: Ten Trillion Zeta Zeros
		I've been watching the progress of Zetagrid.net, waiting for the trillionth nontrivial zero of the Zeta function to be calculated.
		As I write, they have calculated "935.7 billion nontrivial zeros of the Riemann zeta function in 1146 days." I thought I'd have a nice little story next month once they went over a trillion.
		He used this extension of the Zeta function to describe the distribution of primes.
	www.maa.org /editorial/mathgames/mathgames_10_18_04.html (1046 words)

	ZetaGrid - Verification of the Riemann Hypothesis
		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).
		Therefore, I invite all interested people to participate in the verification of the zeros of the Riemann zeta function for a new record.
		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)

	Math 248B - Algebraic Number Theory - Winter 2004
		The Theory of the Riemann Zeta Function, by E.C. Titchmarsh.
		Titchmarsh, "Theory of the Riemann Zeta Function," Chs.
		Notes on the continuation for the Dedekind Zeta Function (last revised: Friday, February 27, 4:30 pm) -- The first 10 pages have been carefully edited.
	math.stanford.edu /~brubaker/math248b.html (675 words)

	The Riemann Hypothesis
		Riemann derived the functional equation of the Riemann zeta function:
		In 1986 it was shown that the first 1,500,000,001 nontrivial zeros of the Riemann zeta function do indeed have real part one-half [VTW86].
		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)

	On the Characteristic Polynomial of a Random Unitary Matrix (Site not responding. Last check: 2007-11-03)
		The analogy between the characteristic polynomial of a random unitary matrix and Riemann's zeta function was first studied by Keating and Snaith in \cite{KS1}.
		For example, they were able to conjecture the asymptotic form of the moments of the zeta function high up the critical line from a random matrix calculation.
		These are then compared with the zeta function, and a conjecture for the left large deviations of $\log\zeta1/2+\i t$ is proposed.
	www.aimath.org /%7Ehughes/PhD_abstract.html (272 words)

	Zeta function (Site not responding. Last check: 2007-11-03)
		The Zeta function (sometimes called the Riemann Zeta Function) was defined by Euler as
		This allows the zeta function to be defined for real values of z greater than 0 but not 1 as follows.
		A well known (and unproved) hypothesis is called the Reimann Hypothesis is that all values of z for which zeta(z) = 0 are of the form 1/2 * i y.
	astronomy.swin.edu.au /~pbourke/analysis/zetafcn (124 words)

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