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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

Table of contents

Properties

The last two properties show that every Dirichlet character χ is completely multiplicative. One can show that χ(n) is a root of unity whenever n and k are coprime.

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.

L-series

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

<math>L(\chi,s) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s}</math>
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.

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.



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