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Class Number Formula
In number theory, the class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function. General statement of the class number formula We start with the following data: * is a number field. * , where denotes the number of real and complex embeddings, real embeddings of , and is the number of complex embeddings of . * is the Dedekind zeta function of . * is the ideal class, class number, the number of elements in the ideal class group of . * is the regulator (mathematics), regulator of . * is the number of root of unity, roots of unity contained in . * is the discriminant of an algebraic number field, discriminant of the Algebraic extension, extension . Then: :Theorem (Class Number Formula). conditionally convergent, converges absolutely for and extends to a meromorphic function (mathematics), function defined for all complex with only one simple pole at , with residue :: \lim_ (s-1) \zeta_K(s) = \frac This is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss Circle Problem
In mathematics, the Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin and with radius r. This number is approximated by the area of the circle, so the real problem is to accurately bound the error term describing how the number of points differs from the area. The first progress on a solution was made by Carl Friedrich Gauss, hence its name. The problem Consider a circle in \mathbb^2 with center at the origin and radius r\ge 0. Gauss's circle problem asks how many points there are inside this circle of the form (m,n) where m and n are both integers. Since the equation of this circle is given in Cartesian coordinates by x^2+y^2= r^2, the question is equivalently asking how many pairs of integers ''m'' and ''n'' there are such that :m^2+n^2\leq r^2. If the answer for a given r is denoted by N(r) then the following list shows the first few values of N(r) for ''r'' an integer between 0 and 12 followed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Unit (number Theory)
In algebraic number theory, a fundamental unit is a generator (modulo the roots of unity) for the unit group of the ring of integers of a number field, when that group has rank 1 (i.e. when the unit group modulo its torsion subgroup is infinite cyclic). Dirichlet's unit theorem shows that the unit group has rank 1 exactly when the number field is a real quadratic field, a complex cubic field, or a totally imaginary quartic field. When the unit group has rank ≥ 1, a basis of it modulo its torsion is called a fundamental system of units. Some authors use the term fundamental unit to mean any element of a fundamental system of units, not restricting to the case of rank 1 (e.g. ). Real quadratic fields For the real quadratic field K=\mathbf(\sqrt) (with ''d'' square-free), the fundamental unit ε is commonly normalized so that (as a real number). Then it is uniquely characterized as the minimal unit among those that are greater than 1. If Δ denotes the discriminant of ''K'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pell Equation
Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinates, the equation is represented by a hyperbola; solutions occur wherever the curve passes through a point whose ''x'' and ''y'' coordinates are both integers, such as the trivial solution with ''x'' = 1 and ''y'' = 0. Joseph Louis Lagrange proved that, as long as ''n'' is not a perfect square, Pell's equation has infinitely many distinct integer solutions. These solutions may be used to accurately approximate the square root of ''n'' by rational numbers of the form ''x''/''y''. This equation was first studied extensively in India starting with Brahmagupta, who found an integer solution to 92x^2 + 1 = y^2 in his ''Brāhmasphuṭasiddhānta'' circa 628. Bhaskara II in the 12th century and Narayana Pandit i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet L-series
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 absol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Kronecker Symbol
In number theory, the Kronecker symbol, written as \left(\frac an\right) or (a, n), is a generalization of the Jacobi symbol to all integers n. It was introduced by . Definition Let n be a non-zero integer, with prime factorization :n=u \cdot p_1^ \cdots p_k^, where u is a unit (i.e., u=\pm1), and the p_i are primes. Let a be an integer. The Kronecker symbol \left(\frac\right) is defined by : \left(\frac\right) = \left(\frac\right) \prod_^k \left(\frac\right)^. For odd p_i, the number \left(\frac\right) is simply the usual Legendre symbol. This leaves the case when p_i=2. We define \left(\frac\right) by : \left(\frac\right) = \begin 0 & \mboxa\mbox \\ 1 & \mbox a \equiv \pm1 \pmod, \\ -1 & \mbox a \equiv \pm3 \pmod. \end Since it extends the Jacobi symbol, the quantity \left(\frac\right) is simply 1 when u=1. When u=-1, we define it by : \left(\frac\right) = \begin -1 & \mboxa 0. Table of values The following is a table of values of Kronecker symbol \left(\frac\r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Discriminant
In mathematics, a fundamental discriminant ''D'' is an integer invariant in the theory of integral binary quadratic forms. If is a quadratic form with integer coefficients, then is the discriminant of ''Q''(''x'', ''y''). Conversely, every integer ''D'' with is the discriminant of some binary quadratic form with integer coefficients. Thus, all such integers are referred to as ''discriminants'' in this theory. There are explicit congruence conditions that give the set of fundamental discriminants. Specifically, ''D'' is a fundamental discriminant if and only if one of the following statements holds * ''D'' ≡ 1 (mod 4) and is square-free, * ''D'' = 4''m'', where ''m'' ≡ 2 or 3 (mod 4) and ''m'' is square-free. The first ten positive fundamental discriminants are: : 1, 5, 8, 12, 13, 17, 21, 24, 28, 29, 33 . The first ten negative fundamental discriminants are: : −3, −4, −7, −8, −11, −15, −19, −20, −23, −24, −31 . Connection with quadratic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harold Davenport
Harold Davenport FRS (30 October 1907 – 9 June 1969) was an English mathematician, known for his extensive work in number theory. Early life Born on 30 October 1907 in Huncoat, Lancashire, Davenport was educated at Accrington Grammar School, the University of Manchester (graduating in 1927), and Trinity College, Cambridge. He became a research student of John Edensor Littlewood, working on the question of the distribution of quadratic residues. First steps in research The attack on the distribution question leads quickly to problems that are now seen to be special cases of those on local zeta-functions, for the particular case of some special hyperelliptic curves such as Y^2 = X(X-1)(X-2)\ldots (X-k). Bounds for the zeroes of the local zeta-function immediately imply bounds for sums \sum \chi(X(X-1)(X-2)\ldots (X-k)), where χ is the Legendre symbol '' modulo'' a prime number ''p'', and the sum is taken over a complete set of residues mod ''p''. In the light of this co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MathOverflow
MathOverflow is a mathematics question-and-answer (Q&A) website, which serves as an online community of mathematicians. It allows users to ask questions, submit answers, and rate both, all while getting merit points for their activities. It is a part of the Stack Exchange Network. It is primarily for asking questions on mathematics research – i.e. related to unsolved problems and the extension of knowledge of mathematics into areas that are not yet known – and does not welcome requests from non-mathematicians for instruction, for example homework exercises. It does welcome various questions on other topics that might normally be discussed among mathematicians, for example about publishing, refereeing, advising, getting tenure, etc. It is generally inhospitable to questions perceived as tendentious or argumentative. Origin and history The website was started by Berkeley graduate students and postdocs Anton Geraschenko, David Zureick-Brown, and Scott Morrison on 28 Septe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal (ring Theory)
In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group. Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Forms
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a fixed field , such as the real or complex numbers, and one speaks of a quadratic form over . If K=\mathbb R, and the quadratic form takes zero only when all variables are simultaneously zero, then it is a definite quadratic form, otherwise it is an isotropic quadratic form. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal group), differential geometry (Riemannian metric, second fundamental form), differential topology ( intersection forms of four-manifolds), and Lie theory (the Killing form). Quadratic forms are not to be confused with a quadratic equation, which has only one variable and includes terms of degree two or less. A quadratic form is o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
