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qoraj
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Junior
Asked: 2021-03-29T15:06:37+00:00 In: ,

Dirichlet character

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Hello,

I need an example of a Dirichlet character .

Thank you in advance

number theory

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1 Answer

  2. acedstud
    This answer was edited.

    A function \chi(n) from the integers \mathbb{Z} to the complex numbers \mathbb{C} is called a Dirichlet character if it satisfies the following properties.

    1. There exists a positive integer k such that \chi(n) = \chi(n + k) for all integers n. k is the period of the function.
    2. \chi(mn) = \chi(m)\chi(n) for all integers m and n. (\chi(n) is completely multiplicative.)
    3. If gcd(n, k) > 1 then chi(n) = 0; if gcd(n, k) = 1 then \chi(n) \ne 0.
    4. \chi(1)=1

    An example of the Dirichlet characters is characters modulo 1. Any character modulo 1 satisfies the given condition for Dirichlet characters with a period, k = 1.

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