Dirichlet character 

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>

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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