In number theory, a Dirichlet character is a function χ from the positive integers to the complex numbers which has the following properties: Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ...

• There exists a positive integer k such that χ(n) = χ(n + k) for all n. This means that the character is periodic with period k.
• χ(n) = 0 for every n with gcd(n,k) > 1
• χ(mn) = χ(m)χ(n) for all positive integers m and n
• χ(1) = 1

The first and third conditions above are sufficient; ie. a Dirichlet character is any complex-valued function on the natural numbers which is both periodic and completely multiplicative. In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf) of two integers which are not both zero is the largest integer that divides both numbers. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... In number theory, a multiplicative function is an arithmetic function f(n) of the positive integer n with the property that f(1) = 1 and whenever a and b are coprime, then f(ab) = f(a) f(b). ...

Properties

The last two properties show that every Dirichlet character χ is completely multiplicative. One can show that χ(n) is a φ(n)th root of unity whenever n and k are coprime, and where φ(n) is the totient function. A detailed construction of Dirichlet characters starting from the basics of modular arithmetic is given in the article on character groups. In number theory, a multiplicative function is an arithmetic function f(n) of the positive integer n with the property that f(1) = 1 and whenever a and b are coprime, then f(ab) = f(a) f(b). ... In mathematics, the nth roots of unity or de Moivre numbers are all the complex numbers which yield 1 when raised to a given power n. ... Coprime - Wikipedia /**/ @import /skins-1. ... 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. ... Modular arithmetic is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... In mathematics, a character group is the group of representations of a group by complex-valued functions. ...

Examples

An example of a Dirichlet character is the function

This character has period 4.

If p is a prime number, then the function In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ...

where is the Legendre symbol, is a Dirichlet character of period p. The Legendre symbol is used by mathematicians in the area of number theory, particularly in the fields of factorization and quadratic residues. ...

Dirichlet L-series

If χ is a Dirichlet character, one defines its Dirichlet L-series by

where s is a complex number with real part > 1. By analytic continuation, this function can be extended to a meromorphic function on the whole complex plane. In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. ... 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 mathematics, the complex plane is a way of visualising the space of the complex numbers. ...

Dirichlet L-series are straightforward generalizations of the Riemann zeta function and appear prominently in the generalized Riemann hypothesis. 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. ... The Riemann hypothesis is one of the most important conjectures in mathematics. ...

A Dirichlet L-series can be expressed as a linear combination of the Hurwitz zeta function, and thus the study of L-series can be unified through a study of the Hurwitz zeta. In mathematics, the Hurwitz zeta function is one of the many zeta functions. ...

History

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. The extension to holomorphic functions was accomplished by Bernhard Riemann. Peter Gustav Lejeune Dirichlet. ... 1831 was a common year starting on Saturday (see link for calendar). ... In number theory, Dirichlets 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. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... Bernhard Riemann. ...