A Dirichlet character is a periodic multiplicative function from the positive integers to the real numbers. In mathematical notation we would say that a Dirichlet character is a function such that
1. for all positive integers and some integer q, and 2. for all positive integers and .
The smallest such for which property 1 holds is known as the period of . Typically we impose the additional restriction that for all integers such that where is the period of ; with this restriction there are exactly such characters.
The Dirichlet characters with period have been completely classified. They are very useful in number theory.
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