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# Topic: Riemann zeta function

###### In the News (Sat 17 Aug 19)

 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. 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. en.wikipedia.org /wiki/Riemann_zeta_function   (1557 words)

 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. en.wikipedia.org /wiki/Riemann_hypothesis   (1847 words)

 Riemann zeta function Article, Riemannzetafunction Information   (Site not responding. Last check: 2007-11-06) Bernhard Riemann realized that the zeta function can be extended by analytic continuation in a unique way to a holomorphic functionζ(s) defined for all complex numbers s with s ≠ 1. It is this functionthat is the object of the Riemann hypothesis. Although mathematicians regard the Riemann zeta function as being primarily relevant to the "purest" of mathematicaldisciplines, number theory, it also occurs in applied statistics (see Zipf's law and Zipf-Mandelbrot law), physics, and the mathematical theory of musical tuning. www.anoca.org /numbers/prime/riemann_zeta_function.html   (628 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   (785 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 Hypothesis in a Nutshell   (Site not responding. Last check: 2007-11-06) 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)

 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)

 Math Forum - Ask Dr. Math   (Site not responding. Last check: 2007-11-06) The Riemann zeta function is defined as follows: infinity zeta(s) = sum (1/n^s) n=1 or, if you like: zeta(s) = 1/1^s + 1/2^s + 1/3^s + 1/4^s +... This function has an infinite number of zeroes, and except for the trivial zeros at integer negative values, all the rest seem to lie on the line real(s) = 1/2. Riemann's hypothesis is that all are on that line. mathforum.org /library/drmath/view/52831.html   (286 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)

 Statistical mechanics: the Riemann zeta function interpreted as a partition function   (Site not responding. Last check: 2007-11-06) Recall that the Riemann Hypothesis seeks to restrict the location of zeros of the Riemann zeta function to a line in the complex plane. In the theory of the distribution of primes, the fundamental object is the Riemann zeta function. 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   (6952 words)

 Riemann's Zeta Function   (Site not responding. Last check: 2007-11-06) "...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.) Titchmarsh, The Theory of the Riemann Zeta-Function, 2 www.maths.ex.ac.uk /~mwatkins/zeta/zetafn.htm   (475 words)

 The Riemann Hypothesis Riemann noted that his zeta function had trivial zeros at -2, -4, -6,... 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]. primes.utm.edu /notes/rh.html   (452 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)

 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)

 Mollifying The Riemann Zeta-Function (ResearchIndex)   (Site not responding. Last check: 2007-11-06) 10 Mean-value theorems in the theory of the Riemann zeta-functi.. 9 Contributions to the theory of the Riemann zeta-function and.. 1 the Keating-Snaith constant in the theory of the Riemann zet.. citeseer.ist.psu.edu /85889.html   (382 words)

 Amazon.com: Riemann's Zeta Function (Pure and Applied Mathematics; a Series of Monographs and Textbooks, 58): Books   (Site not responding. Last check: 2007-11-06) This book is a study of Bernhard Riemann's epoch-making 8-page paper "On the Number of Primes Less Than a Given Magnitude," and of the subsequent developments in the theory which this paper inaugurated. It includes a translation of Riemann's original paper (On the Number of Primes...) which is very nice and most authors now seem to forget to mention (mainly because of the obscure way in which it was written). It has always seemed to me that the very best modern books on the Riemann Zeta Function, and its applications to analytic number theory, are either written at a vey high or a very low level of mathematical sophistication. www.amazon.com /exec/obidos/tg/detail/-/0122327500?v=glance   (1713 words)

 zeta.html The 10^22-nd zero of the Riemann zeta function, A. A nonlinear equation and its application to nearest neighbor spacings for zeros of the zeta function and eigenvalues of random matrices, P. Forrester and A. Odlyzko, in Organic Mathematics, J. On the distribution of spacings between zeros of the zeta function, A. www.dtc.umn.edu /~odlyzko/doc/zeta.html   (409 words)

 Citebase - The Riemann Zeta Function and Vacuum Spectrum Authors: Joffily, S. A variant for the Hilbert and Polya spectral interpretation of the Riemann zeta function is proposed. Instead of looking for a self-adjoint linear operator H, whose spectrum coincides with the Riemann zeta zeros, we look for the complex poles of the S matrix that are mapped into the critical line in coincidence with the nontrivial Riemann zeroes. The Riemann Zeta Function and Vacuum Spectrum 8 [7] E.C. Titchmarsh, The theory of Riemann Zeta-function 2nd. citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:hep-th/0412217   (564 words)

 index.html   (Site not responding. Last check: 2007-11-06) Andrew Odlyzko: Tables of zeros of the Riemann zeta function The first 100,000 zeros of the Riemann zeta function, accurate to within 3*10^(-9). The first 100 zeros of the Riemann zeta function, accurate to over 1000 decimal places. www.dtc.umn.edu /~odlyzko/zeta_tables   (68 words)

 The Prime Glossary: Riemann zeta function   (Site not responding. Last check: 2007-11-06) Riemann extended the definition of Euler's zeta function Euler’s product definition of this function still holds if the real part of s is greater than one. To help understand the values for other complex numbers, Riemann derived the functional equation of the Riemann zeta function: primes.utm.edu /glossary/page.php?sort=RiemannZetaFunction   (102 words)

 ON QUANTUM THEORETICAL ORIGINS OF NEWTONIAN TIME Nevertheless, the kind of linearization arising from De Witt functional equation defined on the "superspace" of old that speaks of the "wave function of the universe" presents severe logical difficulties with regard to a quantum measurement theory. A ψ function for a particle suddenly appearing at any point of space is equally suddenly determined throughout space: similarly, when a particle suddenly disappears. Hilbert space, it is unitarily equivalent to that representation induced in the Hilbert space of holomorphic functions on CÂ² by the SU(2) rotations in CÂ². graham.main.nc.us /~bhammel/PHYS/newtqtime.html   (15363 words)

 Riemann’s Zeta Function Superb, high-level study of one of the most influential classics in mathematics examines landmark 1859 publication entitled “On the Number of Primes Less Than a Given Magnitude,” and traces developments in theory inspired by it. Topics include Riemann’s main formula, the prime number theorem, the Riemann-Siegel formula, large-scale computations, Fourier analysis, and other related topics. English translation of Riemann’s original document appears in the Appendix. store.doverpublications.com /0486417409.html   (250 words)

 The Dirichlet/Riemann zeta Function   (Site not responding. Last check: 2007-11-06) The Dirichlet zeta function, written ζ(f,s) is the sum of f(n)/n There are some functions, such as f(n) = n factorial, that never converge, no matter the value of s. Zeta is defined at the origin iff f is convergent. www.mathreference.com /lc-z,drz.html   (217 words)

 Amazon.com: The Theory of the Riemann Zeta-Function (Oxford Science Publications): Books: E. C. Titchmarsh,D. R. ...   (Site not responding. Last check: 2007-11-06) The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. The Riemann zeta-function has its origin in the identity expressed by the two formulae where n runs through all integers, and where p runs through all primes. approximate functional equation, indefinitely large values, exponent pairs, critical strip, convexity theorem, circle with centre, integral round, uniformly with respect, partial summation, prime number theorem, analytic continuation, complex zeros, uniformly convergent, inner sum, angular region, zeta function www.amazon.com /exec/obidos/tg/detail/-/0198533691?v=glance   (983 words)

 Riemann Zeta Function   (Site not responding. Last check: 2007-11-06) The Riemann Zeta Function is most simply expressed as follows: Now we can use the right side of equation (2) to rewrite the function f(x) in order to solve for C, the constant of integration. Riemann Zeta Function: the extensive MathWorld encyclopedia entry on the most important of functions people.ucsc.edu /~erowland/zeta.html   (464 words)

 Zeta Function Plotter   (Site not responding. Last check: 2007-11-06) The famous Riemann Hypothesis is that all zeros of this function lie on the line The applet uses the Riemann-Siegel formula for computing values of the zeta function. Hopefully the applet controls are more or less obvious; pass the mouse over the various buttons for a short explanation in the ``messages'' area about their function. web.mala.bc.ca /pughg/RiemannZetaComplex   (296 words)

 Citebase - X-Ray of Riemann zeta-function This allow to illustrate many properties of the zeta function of Riemann. Citation coverage and analysis is incomplete and hit coverage and analysis is both incomplete and noisy. [8] Hutchinson, J. I., On the roots of the Riemann zeta-function, Trans. citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0309433   (580 words)

 Riemann zeta function: Theorems The Riemann hypothesis on the zeros of the zeta-function (0 formulas) The equivalent version of the Riemann hypothesis (1 formula) The probability of a lattice point to be visible (0 formulas) functions.wolfram.com /ZetaFunctionsandPolylogarithms/Zeta/31   (41 words)

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