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# Topic: Dedekind zeta function

 Springer Online Reference Works   (Site not responding. Last check: 2007-10-17) The function was subsequently studied by P.G.L. Dirichlet and also, with extraordinary success, by P.L. Chebyshev [2] in the context of the problem of the distribution of prime numbers. Dedekind's zeta-function is the analogue of Riemann's zeta-function for algebraic number fields, and was introduced by R. Dedekind's zeta-functions are the basis of the modern analytic theory of divisors of algebraic number fields. eom.springer.de /Z/z099260.htm   (3255 words)

 Riemann zeta function - Encyclopedia, History, Geography and Biography 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. 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. www.arikah.net /encyclopedia/Riemann_zeta_function   (2058 words)

 Zeta function - Encyclopedia, History, Geography and Biography The theory of L-functions should in the end contain the theory of zeta functions; an L-function is a potentially 'twisted' kind of zeta function. The Selberg class S is an attempt to define zeta functions axiomatically, so that the properties of the class can be studied, and the members of the class classified. The zeta functions should not be confused with the similar-sounding eta function. www.arikah.net /encyclopedia/Zeta_function   (263 words)

 PlanetMath: Factorization of the Dedekind zeta function of an abelian number field The Dedekind zeta function of an abelian number field factors as a product of Dirichlet L-functions as follows. "Factorization of the Dedekind zeta function of an abelian number field" is owned by alozano. This is version 1 of Factorization of the Dedekind zeta function of an abelian number field, born on 2006-06-20. www.planetmath.org /encyclopedia/FactorizationOfTheDedekindZetaFunctionOfAnAbelianNumberField.html   (221 words)

 Encyclopedia: Dedekind zeta function   (Site not responding. Last check: 2007-10-17) dividing p of In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles fore the function. In complex analysis, a pole of a function is a certain type of simple singularity that behaves like the singularity of f(z) = 1/zn at z = 0; a pole of a function f is a point a such that f(z) approaches infinity as z approaches a. Accordingly, the transition between the pseudo-Anosov and the periodic dynamic regime (in Thurston's terminology) in the case of gravity is being reinterpreted in terms of phase transitions in the Farey spin chain whose partition function is just a ratio of two Riemann zeta functions. www.nationmaster.com /encyclopedia/Dedekind-zeta-function   (1516 words)

 PlanetMath: values of Dedekind zeta functions of real quadratic number fields at negative integers "values of Dedekind zeta functions of real quadratic number fields at negative integers" is owned by alozano. See Also: Factorization of the Dedekind zeta function of an abelian number field This is version 2 of values of Dedekind zeta functions of real quadratic number fields at negative integers, born on 2006-06-20, modified 2006-07-19. www.planetmath.org /encyclopedia/ValuesOfDedekindZetaFunctionsOfRealQuadraticNumberFieldsAtNegativeIntegers.html   (230 words)

 Encyclopedia: Riemann zeta function   (Site not responding. Last check: 2007-10-17) 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. Using this equation one sees immediately that the zeta function is zero at the negative even integers. Gram observed that the real part of the zeta function on the critical line tends to be positive, while the imaginary part alternates more regularly between positive and negative values. www.nationmaster.com /encyclopedia/Riemann-zeta_function   (377 words)

 Dedekind zeta function - Wikipedia, the free encyclopedia There is a functional equation for the Dedekind zeta-function, relating its values at s and 1 − s. For the case in which K is an abelian extension of Q, its Dedekind zeta-function can be written as a product of Dirichlet L-functions. In general if K is a Galois extension of Q with Galois group G, its Dedekind zeta-function has a comparable factorization in terms of Artin L-functions. www.wikipedia.org /wiki/Dedekind_zeta_function   (360 words)

 Generalized Riemann hypothesis - RecipeFacts   (Site not responding. Last check: 2007-10-17) 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. By analytic continuation, this function can be extended to a meromorphic function defined on the whole complex plane. The Dedekind zeta function satisfies a functional equation and can be extended by analytic continuation to the whole complex plane. www.recipeland.com /encyclopaedia/index.php/Extended_Riemann_hypothesis   (711 words)

 PlanetMath: Dedekind zeta function Factorization of the Dedekind zeta function of an abelian number field Cross-references: Riemann zeta function, prime ideals, Euler product, simple pole, plane, meromorphic continuation, converges, norm, ideals, ranges, series, analytic continuation, ring of integers, number field This is version 6 of Dedekind zeta function, born on 2002-12-27, modified 2005-03-10. planetmath.org /encyclopedia/DedekindZetaFunction.html   (145 words)

 PlanetMath: values of the Dedekind zeta function of $\mathbb{Q}(\sqrt{5})$ at negative integers PlanetMath: values of the Dedekind zeta function of $\mathbb{Q}(\sqrt{5})$ at negative integers As an example of the parent entry, we exhibit a few values of the Dedekind zeta function This is version 2 of values of the Dedekind zeta function of www.planetmath.org /encyclopedia/ValuesOfTheDedekindZetaFunctionOfMathbbQsqrt5AtNegativeIntegers.html   (100 words)

 What is Riemann zeta function? : Abaara fun facts and uncommon knowledge   (Site not responding. Last check: 2007-10-17) In mathematics, the Riemann zeta function is a It is this function that is the object of the The Riemann zeta function is meromorphic with a single pole of order one at s=1. www.abaara.com /pac/Riemann_zeta_function   (1177 words)

 Advanced Examples function may be used to define the Dedekind zeta function of a number field K. This is essentially what the function Finally, the Dedekind zeta function has a pole at s=1 and we need its residue (or, rather, the residue of zeta^ * (s)) which we compute using the class number formula. The resulting L-function does not satisfy the required functional equation, so something unusual does happen at a prime where both E and K have bad reduction, which in this case must be either p=2 or p=3. magma.maths.usyd.edu.au /magma/htmlhelp/text1430.htm   (1441 words)

 Functional equation (L-function) - Psychology Central   (Site not responding. Last check: 2007-10-17) Therefore use of the functional equation is basic, in order to study the zeta-function in the whole complex plane. Just the same shape of functional equation holds for the Dedekind zeta function of a number field K, with an appropriate gamma-factor that depends only the embeddings of K (in algebraic terms, on the tensor product of K with the real field). There are also functional equations for the local zeta-functions, arising at a fundamental level for the (analogue of) Poincaré duality in étale cohomology. www.grohol.com /psypsych/Functional_equation_%28L-function%29   (613 words)

 ComputeL - Pari package to compute motivic L-functions Dirichlet L-functions of Dirichlet characters and Dedekind zeta-function of a number field Determine the residues at the poles of L Maciej Radziejewski's page Computing zeros of Hecke zeta functions. maths.dur.ac.uk /~dma0td/computel   (514 words)

 [No title] The \emph{Dedekind zeta function} $\zeta_K(s)$ is a function on the complex plane given, for $\Real(s) > 1$, by the absolutely convergent product and sum $\zeta_K(s) = \prod_\gothp (1 - \Norm(\gothp)^{-s})^{-1} = \zeta_K(s) = \sum_{\gotha} \Norm(\gotha)^{-s},$ where in the sum $\gotha$ runs over the nonzero ideals of $\gotho_K$. There is also a functional equation relating the values of $\zeta_K$ at $s$ and $1-s$, and an extended Riemann hypothesis: aside from trivial'' zeros along the negative real axis, the zeroes of $\zeta_K$ all have real part $1/2$. Extend $\chi_\gothm$ to a function on all ideals of $K$ by declaring its value to be 0 on ideals not coprime to $\gothm$. math.mit.edu /~kedlaya/Math254B/zeta.tex   (870 words)

 Dedekind zeta function - Psychology Central   (Site not responding. Last check: 2007-10-17) In mathematics, the Dedekind zeta function is a Dirichlet series defined for any algebraic number field $K$, and denoted $\zeta_K \left(s\right)$ where $s$ is a complex variable. The properties of $\zeta_K\left(s\right)$ as a meromorphic function turn out to be of considerable significance in algebraic number theory. There is a functional equation for the Dedekind zeta function, relating its values at $s$ and $1 - s$. www.grohol.com /psypsych/Dedekind_zeta_function   (356 words)

 Read about Zeta function at WorldVillage Encyclopedia. Research Zeta function and learn about Zeta function here!   (Site not responding. Last check: 2007-10-17) It is widely believed by mathematicians that there is a vast generalization that will tie much of the theory of zeta functions and Dirichlet series together; but the nature of such a general theory is not known. This is a disambiguation page which serves to distinguish between several terms that share a common name. encyclopedia.worldvillage.com /s/b/Zeta_functions   (233 words)

 Erich Hecke - Wikipedia, the free encyclopedia He was born in Buk, Posen Germany (now Poznan Poland), and died in Copenhagen, Denmark. His early work included establishing the functional equation for the Dedekind zeta function, with a proof based on theta functions. The method extended to the L-functions associated to a class of characters now known as Hecke characters or idele class characters: such L-functions are now known as Hecke L-functions. www.wikipedia.org /wiki/Erich_Hecke   (160 words)

 Weierstrass zeta function - All About All   (Site not responding. Last check: 2007-10-17) In mathematics, the Weierstrass functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function Note that the Weierstrass zeta function is basically the logarithmic derivative of the sigma function. The Weierstrass eta function should not be confused with the Dedekind eta function. www.allaboutall.info /catalog/Weierstrass_zeta_function   (208 words)

 Dedekind zeta function   (Site not responding. Last check: 2007-10-17) In mathematics, the Dedekind zeta function is a Dirichlet series defined for any algebraic number field K as a meromorphic function turn out to be of considerable significance in algebraic number theory. There is a functional equation for the Dedekind zeta function, relating its values at s www.abitabouteverything.com /files/d/de/dedekind_zeta_function.html   (243 words)

 Analytic Number Theory course/college Analytische Getaltheorie, Leiden Riemann viewed ζ(s) as a function in the complex variable s, and showed that it has an analytic continuation to C\{1}, with a simple pole with residue 1 at s=1. It is an easy consequence of this functional equation, that ζ(s) has simple zeros in all even negative integers -2,-4,-6,... Their proofs were based on complex analysis, applied to the Riemann zeta function. www.math.leidenuniv.nl /~evertse/ant.shtml   (1203 words)

 CMS Summer 2002 Meeting Multiple zeta values are values of multiple polylogarithms at positive integer arguments. We will discuss generalizations of the classical Hurwitz zeta functions, obtain analytic continuations for them and show that certain special values (namely at negative integers) are given by generalized Bernoulli polynomials. This talk will consider the distribution of nontrivial zeros close to the real axis of the family of all Dedekind Zeta functions of quadratic extensions of a given imaginary quadratic number field. www.cms.math.ca /Events/summer02/abs/nt.html   (1581 words)

 AIF : Tome 53 fascicle 1 -- 2003 The function is a meromorphic function of two complex variables with polar divisor has similar asymptotics to the Riemann zeta function. This phenomenon is associated to a positive convolution semigroup with parameter annalif.ujf-grenoble.fr /Vol53/E531_1/E531_1.html   (165 words)

 The Dedekind Zeta Function   (Site not responding. Last check: 2007-10-17) We shall show that the series on the left hand side of (5.2) is convergent, and defines a continuous function of the real variable The Riemann Zeta function has many applications, most notably in the proof of the Prime Number Theorem and the associated Riemann Hypothesis. By the property of the Riemann Zeta function, given in (5.8) and the fact that ssh.houseofrock.org.uk /m4r/node13.html   (256 words)

 [No title]   (Site not responding. Last check: 2007-10-17) For the fall quarter Math 234A, we shall cover the most basic parts of the theory of zeta functions. This includes the Riemann zeta function, the Dedekind zeta function of a general number field, and the zeta function of a general algerbaic variety over a finite field (Dwork's theorem). If time permits, we shall also discuss the application of the algorithmic theory of zeta functions over finite fields to cryptography. www.math.uci.edu /~dwan/234a   (268 words)

 Computational Strategies for the Riemann Zeta Function - Borwein, Bradley, Crandall (ResearchIndex)   (Site not responding. Last check: 2007-10-17) We provide a compendium of evaluation methods for the Riemann zeta function, presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new. We concentrate primarily on practical computational issues, such issues depending on the domain of the argument, the desired speed of computation, and the level of what we call recycling. 3 An Efficient Algorithm for the Riemann Zeta Function - Borwein, Borwein et al. citeseer.lcs.mit.edu /borwein99computational.html   (686 words)

 What is Generalized Riemann hypothesis? : Abaara fun facts and uncommon knowledge L-functions, which are formally similar to the Riemann zeta function. arithmetic function χ such that there exists a positive integer k with χ(n + k) = χ(n) for all n and χ(n) = 0 whenever gcd(n, k) > 1. The Dedekind zeta function satisfies a functional equation and can be extended by info.abaara.com /pac/Generalized_Riemann_hypothesis   (657 words)

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