Dirichlet Character
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 Dirichlet Character In analytic number theory and related branches of mathematics, a complex-valued arithmetic function \chi:\mathbb\rightarrow\mathbb is a Dirichlet character of modulus m (where m is a positive integer) if for all integers a and b: :1)   \chi(ab) = \chi(a)\chi(b);   i.e. \chi is completely multiplicative. :2)   \chi(a) \begin =0 &\text\; \gcd(a,m)>1\\ \ne 0&\text\;\gcd(a,m)=1. \end (gcd is the greatest common divisor) :3)   \chi(a + m) = \chi(a); i.e. \chi is periodic with period m. The simplest possible character, called the principal character, usually denoted \chi_0, (see Notation below) exists for all moduli: : \chi_0(a)= \begin 0 &\text\; \gcd(a,m)>1\\ 1 &\text\;\gcd(a,m)=1. \end The German mathematician Peter Gustav Lejeune Dirichlet—for whom the character is named—introduced these functions in his 1837 paper on primes in arithmetic progressions. Notation \phi(n) is Euler's totient function. \zeta_n is a complex primitive n-th root of unity: ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Analytic Number Theory In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet ''L''-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem). Branches of analytic number theory Analytic number theory can be split up into two major parts, divided more by the type of problems they attempt to solve than fundamental differences in technique. *Multiplicative number theory deals with the distribution of the prime numbers, such as estimating the number of primes in an interval, and includes the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions. *Additive number the ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος ''isos'' "equal", and μορφή ''morphe'' "form" or "shape". The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are . An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as it is the case for solutions of a univ ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Order (group Theory) In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is ''infinite''. The ''order'' of an element of a group (also called period length or period) is the order of the subgroup generated by the element. If the group operation is denoted as a multiplication, the order of an element of a group, is thus the smallest positive integer such that , where denotes the identity element of the group, and denotes the product of copies of . If no such exists, the order of is infinite. The order of a group is denoted by or , and the order of an element is denoted by or , instead of \operatorname(\langle a\rangle), where the brackets denote the generated group. Lagrange's theorem states that for any subgroup of a finite group , the order of the subgroup divides the order of the group; that is, is a divisor of . In particular, the order of any element is a divisor of . Example The symmetric group S3 has t ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Modular Form In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory. A modular function is a function that is invariant with respect to the modular group, but without the condition that be holomorphic in the upper half-plane (among other requirements). Instead, modular functions are meromorphic (that is, they are holomorphic on the complement of a set of isolated points, which are poles of the function). Modular form theory is a special case of the more general theory of automorphic forms which are functions defined on Lie groups which transform nicely ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Dirichlet L-function In mathematics, a Dirichlet ''L''-series is a function of the form :L(s,\chi) = \sum_^\infty \frac. where \chi is a Dirichlet character and ''s'' a complex variable with real part greater than 1. It is a special case of a Dirichlet series. By analytic continuation, it can be extended to a meromorphic function on the whole complex plane, and is then called a Dirichlet ''L''-function and also denoted ''L''(''s'', ''χ''). These functions are named after Peter Gustav Lejeune Dirichlet who introduced them in to prove the theorem on primes in arithmetic progressions that also bears his name. In the course of the proof, Dirichlet shows that is non-zero at ''s'' = 1. Moreover, if ''χ'' is principal, then the corresponding Dirichlet ''L''-function has a simple pole at ''s'' = 1. Otherwise, the ''L''-function is entire. Euler product Since a Dirichlet character ''χ'' is completely multiplicative, its ''L''-function can also be written as an Euler product in the half-plane of a ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Fourier Transform On Finite Groups In mathematics, the Fourier transform on finite groups is a generalization of the discrete Fourier transform from cyclic to arbitrary finite groups. Definitions The Fourier transform of a function f : G \to \Complex at a representation \varrho : G \to \mathrm(d_\varrho, \Complex) of G is \widehat(\varrho) = \sum_ f(a) \varrho(a). For each representation \varrho of G, \widehat(\varrho) is a d_\varrho \times d_\varrho matrix, where d_\varrho is the degree of \varrho. The inverse Fourier transform at an element a of G is given by f(a) = \frac \sum_i d_ \text\left(\varrho_i(a^)\widehat(\varrho_i)\right). Properties Transform of a convolution The convolution of two functions f, g : G \to \mathbb is defined as (f \ast g)(a) = \sum_ f\!\left(ab^\right) g(b). The Fourier transform of a convolution at any representation \varrho of G is given by \widehat(\varrho) = \hat(\varrho)\hat(\varrho). Plancherel formula For functions f, g : G \to \mathbb, the Plancherel formula s ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Indicator Function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\in A, and \mathbf_(x)=0 otherwise, where \mathbf_A is a common notation for the indicator function. Other common notations are I_A, and \chi_A. The indicator function of is the Iverson bracket of the property of belonging to ; that is, :\mathbf_(x)= \in A For example, the Dirichlet function is the indicator function of the rational numbers as a subset of the real numbers. Definition The indicator function of a subset of a set is a function \mathbf_A \colon X \to \ defined as \mathbf_A(x) := \begin 1 ~&\text~ x \in A~, \\ 0 ~&\text~ x \notin A~. \end The Iverson bracket provides the equivalent notation, \in A/math> or to be used instead of \mathbf_(x)\,. The function \mathbf_A is sometimes denoted , , , or even just . Nota ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Group (mathematics) In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. These three axioms hold for number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the obj ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Relation To Group Characters Relation or relations may refer to: General uses *International relations, the study of interconnection of politics, economics, and law on a global level *Interpersonal relationship, association or acquaintance between two or more people *Public relations, managing the spread of information to the public *Sexual relations, or human sexual activity *Social relation, in social science, any social interaction between two or more individuals Logic and philosophy *Relation (philosophy), links between properties of an object *Relational theory, framework to understand reality or a physical system Mathematics A finitary or ''n''-ary relation is a set of ''n''-tuples. Specific types of relations include: *Relation (mathematics) *Binary relation (or correspondence, dyadic relation, or 2-place relation) *Equivalence relation *Homogeneous relation *Reflexive relation * Serial relation *Ternary relation (or triadic, 3-adic, 3-ary, 3-dimensional, or 3-place relation) Relation may also re ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Chinese Remainder Theorem In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of these integers, under the condition that the divisors are pairwise coprime (no two divisors share a common factor other than 1). For example, if we know that the remainder of ''n'' divided by 3 is 2, the remainder of ''n'' divided by 5 is 3, and the remainder of ''n'' divided by 7 is 2, then without knowing the value of ''n'', we can determine that the remainder of ''n'' divided by 105 (the product of 3, 5, and 7) is 23. Importantly, this tells us that if ''n'' is a natural number less than 105, then 23 is the only possible value of ''n''. The earliest known statement of the theorem is by the Chinese mathematician Sun-tzu in the '' Sun-tzu Suan-ching'' in the 3rd century CE. The Chinese remainder theorem is widely used for computing with la ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Primitive Root Modulo N In modular arithmetic, a number is a primitive root modulo  if every number coprime to is congruent to a power of modulo . That is, is a ''primitive root modulo''  if for every integer coprime to , there is some integer for which ≡ (mod ). Such a value is called the index or discrete logarithm of to the base modulo . So is a ''primitive root modulo''  if and only if is a generator of the multiplicative group of integers modulo . Gauss defined primitive roots in Article 57 of the ''Disquisitiones Arithmeticae'' (1801), where he credited Euler with coining the term. In Article 56 he stated that Lambert and Euler knew of them, but he was the first to rigorously demonstrate that primitive roots exist for a prime . In fact, the ''Disquisitiones'' contains two proofs: The one in Article 54 is a nonconstructive existence proof, while the proof in Article 55 is constructive. Elementary example The number 3 is a primitive root modulo 7 because ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Complex Conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - bi. The complex conjugate of z is often denoted as \overline or z^*. In polar form, the conjugate of r e^ is r e^. This can be shown using Euler's formula. The product of a complex number and its conjugate is a real number: a^2 + b^2 (or r^2 in polar coordinates). If a root of a univariate polynomial with real coefficients is complex, then its complex conjugate is also a root. Notation The complex conjugate of a complex number z is written as \overline z or z^*. The first notation, a vinculum, avoids confusion with the notation for the conjugate transpose of a matrix, which can be thought of as a generalization of the complex conjugate. The second is preferred in physics, where dagger (†) is used for the conjugate ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]