In mathematics, there are two notable Artin conjectures, the legacy of Emil Artin. Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... Emil Artin (March 3, 1898-December 20, 1962) was a mathematician born in Vienna, Austria who spent his career in Germany (mainly in Hamburg) until the Nazi threat when he emigrated to the USA in 1937 where he was at Indiana University 1938-1946, and Princeton University 1946-1958. ...

The first of those concerns the region of the complex plane in which an Artin L-function is an analytic function. Let G be a Galois group of a finite Galois extension L/K of number fields; and let ρ be a group representation of G on a complex vector space of finite dimension. Then the Artin conjecture states that the Artin L-function The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... In mathematics, an analytic function is one that is locally given by a convergent power series. ... In mathematics, a Galois group is a group associated with a certain type of field extension. ... In mathematics, a Galois extension is a field extension that has a Galois group. ... In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days... Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... The fundamental concept in linear algebra is that of a vector space or linear space. ...

L(ρ,s)

is meromorphic in the whole of the complex plane, having at most a pole at s = 1. Further, the multiplicity of the pole will be the multiplicity of the trivial representation in ρ. This article is about the mathematical term; Multiplicity is also the title of a 1996 film. ... In mathematics, in particular group representation theory, a group representation of the group G is called a trivial representation if (i) it is defined on a one-dimensional vector space V over a field K and (ii) all elements g of G act on V as the identity mapping. ...

This is known for one-dimensional representations — the L-functions being then associated to Hecke characters — and in particular for Dirichlet L-functions. Further cases depend on the structure of G, when it is not an abelian group. Those seem to lie quite deep, for example in the work of Tunnell. In number theory, a Dirichlet character is a function χ from the positive integers to the complex numbers which has the following properties: There exists a positive integer k such that χ(n) = χ(n + k) for all n. ... In mathematics, an abelian group is a commutative group, i. ...

What is known in general comes out of Brauer's theorem on induced characters, which was in fact motivated by this application. It tells us, expressed in one kind of language, that the Q-module in the multiplicative group of non-zero meromorphic functions in the right half-plane Re(s) > 1 generated by the Hecke L-functions contains all the Artin L-functions. Here multiplication by 1/k means extraction of a k-th root of an analytic function; which is not a problem away from zeroes of the function, which we know do not occur in that half-plane. If there are zeroes, though, we may need branch cuts. In mathematics, multiplicative group in group theory may mean any group G written in multiplicative notation (rather than additive notation for an abelian group) for its binary operation or in particular the multiplicative group of a field F, namely F{0} under multiplication, written F* or Fx. ... A meromorphic function is a function that is holomorphic on an open subset of the complex number plane C (or on some other connected Riemann surface) except at points in a set of isolated poles, which are certain well-behaved singularities. ... In complex analysis, a branch point may be thought of informally as a point z0 at which a multiple_valued function changes values when one winds once around z0. ...

Therefore the Artin conjecture is concerned with zeroes of L-functions, just as the Riemann hypothesis family of conjectures is. It is believed that it would follow from strong enough results from the Langlands philosophy, relating to the L-functions associated to automorphic representations for GL(n) for all n ≥ 1. In fact this is a folk-theorem; it certainly represents one of the major motivations for the generality present in Langlands' work. RH directs here. ... In mathematics, the Langlands program is a web of far-reaching and influential conjectures that connect number theory and the representation theory of certain groups. ... In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms. ... In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ... As the term is understood by mathematicians, folk mathematics or mathematical folklore means theorems, definitions, proofs, or mathematical facts or techniques that circulate among mathematicians by word-of-mouth but have not appeared in print, either in books or in scholarly journals. ...

The second Artin conjecture relates to the density of the set of primes p modulo which a given integer a > 1 is a primitive root, when a is not a perfect square. For example, take a = 2. It claims that the set of primes p for which 2 is a primitive root has a density C, which is also (in fact) the heuristic 'probability of being a primitive root', namely the rate of growth of the sum Modular arithmetic is a system of arithmetic for integers, sometimes referred to as clock arithmetic, where numbers wrap around after they reach a certain value (the modulus). ... In mathematics, a primitive root may mean either a primitive root modulo n in modular arithmetic, or a primitive n-th root of unity amongst the solutions of xn = 1 in a field. ... The term perfect square is used in mathematics in two meanings: a positive integer which is the square of some other integer, i. ...

Σ φ(p − 1)

summed over primes p up to X, and divided by X/logX to take an average. A more computable definition of C as an infinite product is given. Here φ(m) is Euler's totient function. For a sequence of numbers a1, a2, a3, ... we define the infinite product Π an = a1a2a3. ... In number theory, the totient φ(n) of a positive integer n is defined to be the number of positive integers less than or equal to n and coprime to n. ...

Hooley proved that the second conjecture is a consequence of the first (a conditional proof). He assumed the regularity of L-functions for certain extensions built by Kummer theory, by adjoining k-th roots of unity and the k-th root of a to the rational numbers. Their Galois groups over the rational field Q are not abelian as soon as k > 2. Conditional proof is a proof that takes the form of asserting a conditional, and proving that the premise or antecedent of the conditional necessarily leads to the conclusion. ... In mathematics, a Kummer extension of fields is a field extension L/K where for some given integer n > 1 we have [L:K] = n and L is generated over K by a root of a polynomial Xn − a with a in K, and K contains n distinct roots of... In mathematics, the n-th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ... In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...

Using sieve methods, Roger Heath-Brown showed unconditionally that for all but at most two exceptional prime numbers q there are at least Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. ...

cX/(logX)²

prime numbers p < X, such that q is a primitive root modulo p. This result is not constructive, as far as the exceptions go; it is of course conjectured that there are no exceptions. For historical reasons and in order to have application to the solution of Diophantine equations, results in number theory have been scrutinised more than in other branches of mathematics to see if their content is effectively computable. ...

See also: Brown-Zassenhaus conjecture.