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Topic: Dirichlet character

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Dirichlet series

Johann Peter Gustav Lejeune Dirichlet

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	Dirichlet character - Wikipedia, the free encyclopedia
		A detailed construction of Dirichlet characters starting from the basics of modular arithmetic is given in the article on character groups.
		Dirichlet L-series are straightforward generalizations of the Riemann zeta function and appear prominently in the generalized Riemann hypothesis.
		Dirichlet characters and their L-series were introduced by Dirichlet, in 1831, in order to prove Dirichlet's theorem about the infinitude of primes in arithmetic progressions.
	en.wikipedia.org /wiki/Dirichlet_character (333 words)

	Character (mathematics) - Wikipedia, the free encyclopedia
		There are several meanings of the word character in mathematics, although all are related to the idea of using complex numbers to study a more abstract algebraic structure.
		Dirichlet characters can be seen a special case of this definition.
		If f is a finite-dimensional representation of a group G, then the character of the representation is the function from G to the complex numbers given by the trace of f.
	en.wikipedia.org /wiki/Character_(mathematics) (233 words)

	PlanetMath: character of a finite group (Site not responding. Last check: 2007-11-07)
		"character of a finite group" is owned by alozano.
		Cross-references: Dirichlet characters, multiplicative group, group homomorphism, field, finite group
		This is version 3 of character of a finite group, born on 2004-02-20, modified 2005-04-25.
	planetmath.org /encyclopedia/Character2.html (68 words)

	Dirichlet (Site not responding. Last check: 2007-11-07)
		Dirichlet did not remain in Rome for the whole period, but visited Sicily and then spent the winter of 1844/45 in Florence before returning to Berlin in the spring of 1845.
		Dirichlet had a high teaching load at the University of Berlin, being also required to teach in the Military College and in 1853 he complained in a letter to his pupil Kronecker that he had thirteen lectures a week to give in addition to many other duties.
		Dirichlet is also well known for his papers on conditions for the convergence of trigonometric series and the use of the series to represent arbitrary functions.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Dirichlet.html (2043 words)

	Character Group Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-11-07)
		In mathematics, a character group is the group of representations of a group by complex-valued functions.
		The characters of an irreducible representation are orthogonal.
		The primary importance of the character group is in number theory, where it is used to construct Dirichlet characters.
	www.karr.net /search/encyclopedia/Character_group (670 words)

	Generalized Riemann hypothesis - Wikipedia, the free encyclopedia
		A Dirichlet character is a completely multiplicative 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 generalized Riemann hypothesis asserts that for every Dirichlet character χ and every complex number s with L(χ,s) = 0: if the real part of s is between 0 and 1, then it is actually 1/2.
		Dirichlet's theorem states that if a and d are coprime, then such an arithmetic progression contains infinitely many prime numbers.
	en.wikipedia.org /wiki/Generalized_Riemann_hypothesis (698 words)

	PlanetMath: Dirichlet character (Site not responding. Last check: 2007-11-07)
		Dirichlet characters are usually denoted by the Greek letter
		A character which is not induced by any other character is called primitive.
		This is version 6 of Dirichlet character, born on 2003-01-20, modified 2004-03-28.
	planetmath.org /encyclopedia/DirichletCharacter.html (109 words)

	Dirichlet characters (Site not responding. Last check: 2007-11-07)
		The first zeros on the critical line of some Dirichlet L-series...
		Johan Bosman, Lenny Taelman On sums of sums of values of Dirichlet characters NA...
		Bloch-Kato conjecture and main conjecture of Iwasawa theory for Dirichlet charac...
	www.scienceoxygen.com /math/324.html (140 words)

	NSDL Metadata Record -- Dirichlet character
		Dirichlet character modulo m is a group homomorphism from...
		Dirichlet characters are usually denoted by the Greek letter chi.
		A character which is not induced by any other character is called...
	nsdl.org /mr/1032580 (151 words)

	Search Results for Dirichlet (Site not responding. Last check: 2007-11-07)
		Many details of the Dirichlet family are given in [6] where it is shown that the Dirichlets came from the neighbourhood of Liege in Belgium and not, as many had claimed, from France.
		Dirichlet was appointed to the Berlin Academy in 1831 and an improving salary from the university put him in a position to marry, and he married Rebecca Mendelssohn, one of the composer Felix Mendelssohn's two sisters.
		In 1859 Dirichlet died and Riemann was appointed to the chair of mathematics at Gottingen on 30 July.
	www-groups.dcs.st-and.ac.uk /history/Search/historysearch.cgi?SUGGESTION=Dirichlet&CONTEXT=1 (5322 words)

	Character: Definition and Links by Encyclopedian.com - All about Character (Site not responding. Last check: 2007-11-07)
		In general, a character is a distinctive significant mark or feature.
		The word originates from the Greek word χαρακτηρ via the Latin word character, an instrument for marking or graving.
		In Catholicism, a supernatural mark made on a person's soul by some sacraments; see sacramental character.
	www.encyclopedian.com /ch/Character.html (233 words)

	Analytic Number Theory
		Dirichlet used arithmetic progressions to show that every arithmetic progression kn+h, where h and k are relatively prime, contains infinitely many primes.
		where c(n) is a Dirichlet character mod k and s is a real number greater than 1 (or a complex number with real part greater than 1).
		This is called a Dirichlet series with coefficients f(n), and the function F(s) is called a generating function of the coefficients.
	www.risberg.ws /Hypertextbooks/Mathematics/Numbers/analytic.htm (724 words)

	Creation Functions (Site not responding. Last check: 2007-11-07)
		The group of Dirichlet characters mod N with image in the order-r cyclic subgroup of the ring R generated by the root of unity z.
		This is a Dirichlet character of modulus equal to the least common multiple of the moduli of x and y.
		KroneckerCharacter(D) is the quadratic Dirichlet character corresponding to the quadratic field Q(Sqrt(D)).
	www.math.niu.edu /help/math/magmahelp/text1052.html (2009 words)

	Encyclopedia: Dirichlet's theorem (Site not responding. Last check: 2007-11-07)
		In number theory, Dirichlet's theorem states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n > 0, or in other words: there are infinitely many primes which are congruent to a modulo d.
		The theorem in the above form was first conjectured by Gauss and proved by Dirichlet in 1835 with Dirichlet L-series.
		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.
	www.nationmaster.com /encyclopedia/Dirichlet%27s-theorem (960 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Basic properties of Riemann's zeta-function and Dirichlet's $L$-functions are surveyed; it may be a bit misleading for a non-specialist that the zero-free region for the zeta-function (in Theorem 8.6) is not given in the sharpest known form.
		The Riemann hypothesis can also be formulated and verified for related functions like Dirichlet $L$-functions (important for the study of the distribution of primes in arithmetic progressions), and the Epstein zeta function (important for the study of quadratic forms and number fields).
		The method consists in evaluating Dirichlet series, and several of their derivatives, at a set of regularly spaced values; this is done by using the fast Fourier transform to reduce the problem to the evaluation of rational functions.
	www.math.niu.edu /~rusin/known-math/99/zeta (3655 words)

	Generating function - Wikipedia, the free encyclopedia
		There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series; definitions and examples are given below.
		Dirichlet series are often classified as generating functions, although they are not strictly formal power series.
		is a Dirichlet character then its Dirichlet series generating function is called a Dirichlet L-series.
	en.wikipedia.org /wiki/Generating_function (600 words)

	[No title]
		Input, character (len = *) STRING, a string to be examined.
		character c integer i integer ierror integer isgn integer istate integer ival integer last integer lens character (len = *) string !
		characters such as a trailing comma or blanks.
	orion.math.iastate.edu /burkardt/b_src/bdmlib/bdmlib.f90 (1819 words)

	Introduction
		A Dirichlet character is a homomorphism varepsilon:(Z/NZ)^ * -> C^ * of abelian groups.
		Dirichlet characters are of interest because they decompose M_k(Gamma_1(N)) into more manageable chunks.
		This shows that there are no cuspidal newforms with Dirichlet character of order 2.
	www.math.lsu.edu /magma/text1301.htm (1519 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		\documentclass[12pt]{article} \begin{document} The Dirichlet L-series associated to a \htmladdnormallink{Dirichlet character}{http://planetmath.org/encyclopedia/DirichletCharacter.html} $\chi$ is the \htmladdnormallink{series}{http://planetmath.org/encyclopedia/Series.html} \begin{equation} L(\chi,s)=\sum_{n=1}^\infty \frac{\chi(n)}{n^s}.
		For non-trivial, primitive characters $\chi$ mod m, $L(\chi,s)$ admits an \htmladdnormallink{analytic}{http://planetmath.org/encyclopedia/Analytic.html} continuation to all of $\mathbb{C}$ and satsfies the \htmladdnormallink{symmetric}{http://planetmath.org/encyclopedia/Symmetric.html} \htmladdnormallink{functional equation}{http://planetmath.org/encyclopedia/FunctionalEquation.html} \begin{equation} L(\chi,s)\left(\frac{m}{\pi}\right)^{s/2}\Gamma\left(\frac{s+e_\chi}{2}\right)=\frac{g_1(\chi)}{i^{e_\chi}\sqrt{m}}L(\chi^{-1},1-s)\left(\frac{m}{\pi}\right)^{\frac{1-s}{2}}\Gamma\left(\frac{1-s+e_\chi}{2}\right).
		Again assuming that $\chi$ is non-trivial and primitive character mod m, if $k$ is a positive \htmladdnormallink{integer}{http://planetmath.org/encyclopedia/Integer.html}, we have \begin{equation} L(\chi,1-k)=-\frac{B_{k,\chi}}{k}, \end{equation} where $B_{k,\chi}$ is a \htmladdnormallink{generalized Bernoulli number}{http://planetmath.org/encyclopedia/GeneralizedBernoulliNumber.html}.
	www.ma.utexas.edu /~jcorneli/e/work%20folder/FEM-2004-08-16/TeX/11M06--DirichletLSeries.tex (337 words)

	Creation and Basic Functions
		If eps is a Dirichlet character and k >= 2 is an integer, let S be the space of modular forms with weight k and character a Galois conjugate of eps.
		Commands are available which can retrieve the base ring, dimension, character of the defining modular form, a field of definition, the level, the sign, the weights, and a short name of a modular abelian variety.
		If f is a newform then an inner twist of f is a Dirichlet character chi such that the twist of f by chi equals a Galois conjugate of f, at least at Fourier coefficients whose subscript is coprime to some fixed integer.
	magma.maths.usyd.edu.au /magma/htmlhelp/text1321.htm (5832 words)

	POISSON - LoveToKnow Article on POISSON (Site not responding. Last check: 2007-11-07)
		Throughout the empire Poisson faithfully adhered to the family principles, and refused to worship Napoleon.
		When the Bourbons were restored, his hatred against Napoleon led him to become a Legitimista conclusion which says more for the simplicity of his character than for the strength or logic of his political creed.
		He was faithful to the Bourbons during the Hundred Days; in fact, was L This prediction is sometimes attributed to Laplace.
	97.1911encyclopedia.org /P/PO/POISSON.htm (1392 words)

	No Title (Site not responding. Last check: 2007-11-07)
		The Grand Riemann Hypothesis is the assertion that all critical zeros of Dirichlet
		A consequence of RH is the Lindelöf Hypothesis (LH) which is a bound for
		where the * indicates that the sum is over primitive characters.
	www.aimath.org /projects/siegel/siegel.html (609 words)

	Open Questions: The Riemann Hypothesis
		Dirichlet (Riemann's favorite teacher) very early on considered certain interesting series (the original Dirichlet series) long before there was any suspicion of just how deep the resulting theory might be, let alone of any connection with general questions of algebraic number theory.
		A character (specifically, a "Dirichlet" character) is a complex-valued function defined for integers and having the multiplicative property: if χ is a character, then χ(ab) = χ(a)χ(b).
		Characters can be defined for any group, and indeed they are a fundamental tool in the theory of "group representations", which is a way of studying abstract groups in terms of groups of matrices.
	www.openquestions.com /oq-ma014.htm (14106 words)

	Weekly Events 022403 (Site not responding. Last check: 2007-11-07)
		Let $\tau$ be the primitive Dirichlet character of conductor $4$, let $\chi$ be the primitive even Dirichlet character of conductor $8$ and let $k$ be an integer.
		Then the $U_2$ operator acting on cuspidal overconvergent modular forms of weight $2k+1$ and character $\tau$ has slopes in the arithmetic progression $\left\{2,4,\ldots,2n,\ldots\right\}$, and the $U_2$ operator acting on cuspidal overconvergent modular forms of weight $k$ and character $\chi \cdot \tau^k$ has slopes in the arithmetic progression $\left\{1,2,\ldots,n,\ldots\right\}$.
		The barotropic tide generates internal gravity waves due to sea floor topography, which leads to conversion of tidal energy into smaller scale waves and eventually dissipation.
	www.math.uci.edu /w9.html (617 words)

	Introduction (Site not responding. Last check: 2007-11-07)
		Their generality makes modular symbols a natural tool in applications ranging from verification of modularity of Galois representations to elliptic curve computations.
		A mod N Dirichlet character varepsilon is a homomorphism varepsilon:(Z/NZ)^ * -> F^ *.
		The vector space Mm_k(N, varepsilon;F) of modular symbols of weight k, level N and character varepsilon over F is the quotient of Mm_k by the subspace generated by all x - varepsilon(u)g(x), for x in Mm_k and g=pmatrix(uandv cr wandz) in Gamma_0(N).
	www.umich.edu /~gpcc/scs/magma/text1099.htm (420 words)

	Character (mathematics) Info - Bored Net - Boredom (Site not responding. Last check: 2007-11-07)
		Character (mathematics) Info - Bored Net - Boredom
		If A is an Abelian group, a character is a group homomorphism into the multiplicative group of complex numbers.
		If f is a representation of a group G, then the character of the representation is the function from G to the complex numbers given by the trace of f.
	www.borednet.com /e/n/encyclopedia/c/ch/character__mathematics_.html (134 words)

	[No title]
		GETWGT updates the Dirichlet mixture weights based on a set of counts.
		The data in the file is delimited by keywords.
		Input, character (len = *) S1, S2, the strings to be compared.
	orion.math.iastate.edu /burkardt/b_src/getwgt/getwgt.f90 (2092 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Several improvements have been incorporated: a more direct proof of the meromorphic character of the Eisenstein series, an explicit formula for the translation representations and a simpler derivation of the spectral representations.
		In a paper, de Branges mentioned that his approach to the generalized Riemann hypothesis using Hilbert spaces of entire functions is related to the Lax-Phillips theory of scattering.
		We work with perturbations in characters varieties of $\Gamma_0(q)$ and study the effects on the spectral and scattering theory of the Laplace operator.
	www.maths.ex.ac.uk /~mwatkins/zeta/NTscattering.htm (4844 words)

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