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	Informat.io on Riemann Zeta Function (Site not responding. Last check: 2007-09-18)
		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.
		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.
		Zeta function regularization is used as one possible means of regularization of divergent series in quantum field theory.
	www.informat.io /?title=riemann-zeta-function (1597 words)

	PlanetMath: index of special functions
		The term “special function” is not a completely precise mathematical term.
		It usually refers to a function of one or more real or complex variables which is either of use in some application or interesting in its own right, and hence has been studied enough to warrant giving it a name.
		This is version 22 of index of special functions, born on 2004-10-01, modified 2005-05-15.
	planetmath.org /encyclopedia/IndexOfSpecialFunctions.html (195 words)

	Derivatives of the Hurwitz Zeta Function for Rational Arguments (Site not responding. Last check: 2007-09-18)
		Derivatives of the Hurwitz Zeta Function for Rational Arguments
		The functional equation for the Hurwitz Zeta function $\zeta(s, a)$ is used to obtain formulas for derivatives of $\zeta(s, a)$ at negative odd $s$ and rational $a$.
		For several of these rational arguments, closed form expressions are given in terms of simpler transcendental functions, like the logarithm, the polygamma function, and the Riemann Zeta function.
	www-2.cs.cmu.edu /~adamchik/articles/hurwitz.htm (85 words)

	Search ScienceWorld
		The Hurwitz zeta function zeta(s,a) is a generalization of the Riemann zeta function zeta(s) that is also known as the generalized zeta function.
		The Weierstrass zeta function zeta(z;g_2,g_3) is the quasiperiodic function defined by (dzeta(z;g_2,g_3))/(dz)=-P(z;g_2,g_3), where P(z;g_2,g_3) is the Weierstrass elliptic function with invariants g_2 and g_3, with lim_(z->0)[zeta(z;g_2,g_3)-z^(-1)]==0.
		The prime zeta function P(n)=sum_(p)1/(p^n), where the sum is taken over primes is a generalization of the Riemann zeta function zeta(n)=sum_(k==1)^infty1/(k^n), where the sum is over all integers.
	scienceworld.wolfram.com /search/index.cgi?as_q=zeta (391 words)

	Riemann zeta function (Site not responding. Last check: 2007-09-18)
		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 meromorphic function ζ(''s'') defined for all complex numbers s with s ≠ 1.
		It is this function that is the object of the Riemann hypothesis.
	riemann-zeta-function.kiwiki.homeip.net (1180 words)

	Hurwitz zeta function - Wikipedia, the free encyclopedia
		The Hurwitz zeta function can be extended by analytic continuation to a meromorphic function defined for all complex numbers s with s≠1.
		The functional equation relates values of the zeta on the left- and right-hand sides of the complex plane.
		Note that this latter form is the functional equation for the Riemann zeta function, as originally given by Riemann.
	en.wikipedia.org /wiki/Hurwitz_zeta_function (821 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)

	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)

	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)

	Espresso: specfunc.h File Reference (Site not responding. Last check: 2007-09-18)
		This are the modified Hurwitz zeta function and the modified Bessel functions of first and second kind.
		The Hurwitz zeta function is evaluated using the Euler-MacLaurin summation formula, the Bessel functions are evaluated using several different Chebychev expansions.
		Both achieve a precision of nearly machine precision, which is no problem for the Hurwitz zeta function, which is only used when determining the coefficients for the modified polygamma functions (see mmm-common.h).
	www.mpip-mainz.mpg.de /~arnolda/espresso_doc/specfunc_8h.html (401 words)

	Statistical mechanics: the Riemann zeta function interpreted as a partition function (Site not responding. Last check: 2007-09-18)
		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 (8107 words)

	33: Special functions (Site not responding. Last check: 2007-09-18)
		Special functions are just that: specialized functions beyond the familiar trigonometric or exponential functions.
		Functions with an addition formula (F(x+y)=P(F(x),F(y)) P a polynomial) are elliptic functions
		Formal definition of the sine function (via integrals) and derivation of some of its properties.
	www.mat.niu.edu /~rusin/known-math/index/33-XX.html (611 words)

	SFSU MATH 899: Analytic Number Theory, Fall 2005 (Site not responding. Last check: 2007-09-18)
		Properties of the gamma function; integral representation for the Hurwitz zeta function
		A contour integral representation for the Hurwitz zeta function; the analytic continuation of the Hurwitz zeta function; analytic continuation of ζ(s) and L(s,χ)
		A functional equation for the Hurwitz zeta function; the functional equation for L-functions; evaluation of ζ(–n,a); properties of Bernoulli numbers and Bernoulli polynomials
	userwww.sfsu.edu /~dirt/899/schedule.html (190 words)

	The Multiple Gamma and Related Functions Project
		Artin, The Gamma function, Holt Rineart and Winston, 1964.
		Montgomery, The pair correlation of the zeta function, Proc.
		Odlyzko, The 10^20th zero of the Riemann zeta function and 70 million of its neighbors, Preprint, 1989.
	www.cs.cmu.edu /~adamchik/Barnes/bibliography.html (2891 words)

	Wikinfo \| List of mathematical functions
		Trigonometric functions: sine, cosine, etc.; used in geometry and to describe periodic phenomena.
		Euler's phi function: Number of numbers relatively prime to a given one.
		Images, some of which are used under the doctrine of Fair use or used with permission, may not be available.
	www.wikinfo.org /wiki.php?title=List_of_mathematical_functions (497 words)

	TI-89 BASIC Math Programs - ticalc.org
		Function mac(argument) searches for near 40 elementary Mathematical Constants which can be find in the results of some definite integrals, infinite sums and in other fields of mathematics (for example pi, pi^2, e, gamma, phi, ln(2)...) and for simple fractions in the given argument.
		Supporting functions are the beta, incomplete beta, regularized beta, error function, complementary error function, double factorial, rising factorial, gamma, natural log gamma, lower incomplete gamma, upper incomplete gamma, and regularized gamma.
		This function calculates the center of gravity of a convex polygon given the matrix of vertexes.
	www.ticalc.org /pub/89/basic/math/date.html (6886 words)

	Amazon.ca: The Lerch zeta-function: Books: Ramunas Garunkstis (Site not responding. Last check: 2007-09-18)
		The majority of the presented results are new: on approximate functional equations and its applications and on zero distribution (zero-free regions, number of nontrivial zeros etc).
		Special attention is given to limit theorems in the sense of the weak convergence of probability measures for the Lerch zeta-function.
		From limit theorems in the space of analytic functions the universitality and functional independence is derived.
	www.amazon.ca /Lerch-Zeta-Function-Antanas-Laurincikas/dp/1402010141 (354 words)

	Bernoulli numbers and polynomials
		S.C. Woon, "Analytic continuation of Bernoulli numbers, a new formula for the Riemann zeta function, and the phenomenon of scattering of zeros"
		S.C. Woon, "A new representation of the Riemann zeta function \zeta(s)"
		Kuznetsov, "Nontrivial zeros of the Riemann zeta function as the limit of eigenvalues of nonsymmetric matrices" (preprint 2006)
	www.maths.ex.ac.uk /~mwatkins/zeta/bernoulli.htm (925 words)

	Polygamma function - ExampleProblems.com
		In mathematics, the polygamma function of order m is defined as the m+1 'th derivative of the logarithm of the gamma function:
		This representation can be written more compactly in terms of the Hurwitz zeta function as
		alternately, the Hurwitz zeta can be understood to generalize the polygamma to arbitrary, non-integer order.
	www.exampleproblems.com /wiki/index.php/Polygamma_function (113 words)

	Zipf\'s law (Site not responding. Last check: 2007-09-18)
		The classic case of Zipf's law is a "1/''f'' function".
		The value of this sum is ζ(''s''), where ζ is Riemann's zeta function.
		The "constant" in this case is the reciprocal of the Hurwitz zeta function evaluated at s.
	zipfs-law.iqnaut.net (774 words)

	Hurwitz zeta ... (Site not responding. Last check: 2007-09-18)
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	g.msn.com /9SE/1?http://en.wikipedia.org/wiki/Hurwitz_zeta_function&&DI=6244&IG=7b94efd7fa63463898fb5a5bfd75793e&POS=4&CM=WPU&CE=4&CS=AWP&SR=4 (130 words)

	www.myspace.com/riemannzeta
		Adamchik, V. and Srivastava, H. "Some Series of the Zeta and Related Functions." Analysis 18, 131-144, 1998.
		I'm 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.
		I'm defined over the complex plane for one complex variable, and am conventionally denoted "s" (instead of the usual "z") in deference to the notation used by Riemann in his 1859 paper that founded the study of this function (Riemann 1859).
	www.myspace.com /riemannzeta (262 words)

	11M: Zeta and L-functions: analytic theory
		Estimating the number of primes less than N with zeros of the zeta function.
		Connection between zeros of zeta and the error term in the PNT
		Artin's conjecture on zeta functions of number fields.
	www.math.niu.edu /~rusin/known-math/index/11MXX.html (243 words)

	On "Mean value properties of the Hurwitz zeta-function " (Site not responding. Last check: 2007-09-18)
		Andersson, Johan,, Mean value properties of the Hurwitz zeta-function, Mathematica Scandinavica, 71, 1992,(295-300)
		[1] M. Katsurada, and K. Matsumuto, Explicit formulas and asymptotic expansions for certain mean square of Hurwitz zeta-functions, Proc.
		[3] W.P. Zhang, On the mean square value of the Hurwitz zeta-function, Illinois journ.
	www.math.su.se /~johana/mvph.html (65 words)

	Zipf's law - Wikipedia, the free encyclopedia
		The simplest case of Zipf's law is a "1/f function".
		Indeed, the term Zipf's law sometimes simply means the zeta distribution, since probability distributions are sometimes called "laws".
		In the tail of the Yule-Simon distribution the frequencies are approximately
	en.wikipedia.org /wiki/Zipf%27s_Law (1015 words)

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