Dirichlet character
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. 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.
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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.
Examples
An example of a Dirichlet character is the function χ(n) = (-1)(n-1)/2 for odd n and χ(n) = 0 for even n. This character has period 4.
If p is a prime number, then the function χ(n) = (n/p) (the Legendre symbol) is a Dirichlet character of period p.
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.
Dirichlet L-series are straightforward generalizations of the Riemann zeta function and appear prominently in the generalized Riemann hypothesis.
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.
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.