Editing Dirichlet character (section)

{{Short description|Complex-valued arithmetic function}}
In [[analytic number theory]] and related branches of mathematics, a complex-valued [[arithmetic function]]  <math>\chi:\mathbb{Z}\rightarrow\mathbb{C}</math> is a '''Dirichlet character of modulus <math>m</math>''' (where <math>m</math> is a positive integer) if for all integers <math>a</math> and <math>b</math>:<ref>This is the standard definition; e.g. Davenport p.27; Landau p. 109; Ireland and Rosen p. 253</ref>
# <math>\chi(ab) = \chi(a)\chi(b);</math> that is, <math>\chi</math> is [[Completely multiplicative function|completely multiplicative]].
# <math>
\chi(a)
\begin{cases}
=0 &\text{if } \gcd(a,m)>1\\
\ne 0&\text{if }\gcd(a,m)=1.
\end{cases}</math>  (gcd is the [[greatest common divisor]])
# <math>\chi(a + m) = \chi(a)</math>; that is, <math>\chi</math> is periodic with period <math>m</math>.
The simplest possible character, called the '''principal character''', usually denoted <math>\chi_0</math>, (see [[#Notation|Notation]] below) exists for all moduli:<ref>Note the special case of modulus 1: the unique character mod 1 is the constant 1; all other characters are 0 at 0</ref>
:<math>
\chi_0(a)=
\begin{cases}
0 &\text{if } \gcd(a,m)>1\\
1 &\text{if } \gcd(a,m)=1.
\end{cases}</math>
The German mathematician [[Peter Gustav Lejeune Dirichlet]]—for whom the character is named—introduced these functions in his 1837 paper on [[Dirichlet's theorem on arithmetic progressions|primes in arithmetic progressions]].<ref>Davenport p. 1</ref><ref>An English translation is in External Links</ref>