Editing Dirichlet character (section)

== Notation ==

<math>\phi(n)</math> is [[Euler's totient function]].<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Totient Function |url=https://mathworld.wolfram.com/TotientFunction.html |access-date=2025-02-09 |website=mathworld.wolfram.com |language=en}}</ref>

<math>\zeta_n</math> is a complex primitive [[Root of unity|n-th root of unity]]:
:<math>
\zeta_n^n=1,</math> but <math>\zeta_n\ne 1, \zeta_n^2\ne 1, ... \zeta_n^{n-1}\ne 1.</math>

<math>(\mathbb{Z}/m\mathbb{Z})^\times</math> is the [[Multiplicative group of integers modulo n|group of units mod <math>m</math>]]. It has order <math>\phi(m).</math>

<math>\widehat{(\mathbb{Z}/m\mathbb{Z})^\times}</math> is the group of Dirichlet characters mod <math>m</math>.

<math>p, p_k,</math> etc. are  [[prime number]]s.

<math>(m,n)</math> is a standard<ref>Used in Davenport, Landau, Ireland and Rosen</ref> abbreviation<ref><math>(rs,m)=1</math> is equivalent to <math>\gcd(r,m)=\gcd(s,m)=1</math></ref> for <math>\gcd(m,n)</math>

<math>\chi(a), \chi'(a), \chi_r(a),</math> etc. are Dirichlet characters. (the lowercase [[Chi (letter)|Greek letter chi]] for "character")

There is no standard notation for Dirichlet characters that includes the modulus. In many contexts (such as in the proof of Dirichlet's theorem) the modulus is fixed. In other contexts, such as this article, characters of different moduli appear. Where appropriate this article employs a variation of [https://lmfdb.org/knowledge/show/character.dirichlet.conrey Conrey labeling] (introduced by [[Brian Conrey]] and used by the [https://www.lmfdb.org/ LMFDB]).

In this labeling characters for modulus <math>m</math> are denoted <math>\chi_{m, t}(a)</math> where the index <math>t</math> is described in the section [[#The group of characters|the group of characters]] below. In this labeling, <math>\chi_{m,\_}(a)</math> denotes an unspecified character and
<math>\chi_{m,1}(a)</math> denotes the principal character mod <math>m</math>.