Editing Dirichlet character (section)

== Sufficient conditions ==
It is not necessary to establish the defining  properties 1) – 3) to show that a function is a Dirichlet character.

=== From Davenport's book ===

If <math>\Chi:\mathbb{Z}\rightarrow\mathbb{C}</math> such that
:1) &nbsp; <math>\Chi(ab) = \Chi(a)\Chi(b),</math>
:2) &nbsp; <math>\Chi(a + m) = \Chi(a)</math>,
:3) &nbsp; If <math>\gcd(a,m)>1</math> then <math>\Chi(a)=0</math>, but
:4) &nbsp; <math>\Chi(a)</math> is not always 0,
then <math>\Chi(a)</math> is one of the <math>\phi(m)</math> characters mod <math>m</math><ref>Davenport p. 30</ref>

=== Sárközy's Condition ===
A Dirichlet character is a completely multiplicative function <math>f: \mathbb{N} \rightarrow \mathbb{C}</math> that satisfies a [[linear recurrence relation]]: that is, if <math>a_1 f(n+b_1) + \cdots + a_kf(n+b_k) = 0

</math>

for all positive integers <math>n</math>, where <math>a_1,\ldots,a_k</math> are not all zero and <math>b_1,\ldots,b_k</math> are distinct then <math>f</math> is a Dirichlet character.<ref>Sarkozy</ref>

=== Chudakov's Condition ===
A Dirichlet character is a completely multiplicative function <math>f: \mathbb{N} \rightarrow \mathbb{C}</math> satisfying the following three properties: a) <math>f</math> takes only finitely many values; b) <math>f</math> vanishes at only finitely many primes; c) there is an <math>\alpha \in \mathbb{C}</math> for which the remainder

<math>\left|\sum_{n \leq x} f(n)- \alpha  x\right| </math>

is uniformly bounded, as <math>x \rightarrow \infty</math>. This equivalent definition of Dirichlet characters was conjectured by Chudakov<ref>Chudakov</ref> in 1956, and proved in 2017 by Klurman and Mangerel.<ref>Klurman</ref>