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number > number theory > analytic number theory > L-function > Dirichlet character

mathematical analysis > complex analysis > meromorphic function > L-function > Dirichlet character

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Dirichlet character

In analytic number theory and related branches of mathematics, a complex-valued arithmetic function ${\\displaystyle \\chi :\\mathbb {Z} \ ightarrow \\mathbb {C} }$ ${\\displaystyle \\chi :\\mathbb {Z} \ ightarrow \\mathbb {C} }$ is a Dirichlet character of modulus ${\\displaystyle m}$ (where ${\\displaystyle m}$ $m$ is a positive integer) if for all integers ${\\displaystyle a}$ $a$ and ${\\displaystyle b}$ $b$ :
1. ${\\displaystyle \\chi (ab)=\\chi (a)\\chi (b);}$ that is, ${\\displaystyle \\chi }$ is completely multiplicative.
2. ${\\displaystyle \\chi (a){\egin{cases}=0&{\ ext{if }}\\gcd(a,m)>1\\\\\ eq 0&{\ ext{if }}\\gcd(a,m)=1.\\end{cases}}}$ (gcd is the greatest common divisor)
3. ${\\displaystyle \\chi (a+m)=\\chi (a)}$ ; that is, ${\\displaystyle \\chi }$ is periodic with period ${\\displaystyle m}$ .
The simplest possible character, called the principal character, usually denoted ${\\displaystyle \\chi _{0}}$ $\\chi _{0}$ , (see Notation below) exists for all moduli:
${\\displaystyle \\chi _{0}(a)={\egin{cases}0&{\ ext{if }}\\gcd(a,m)>1\\\\1&{\ ext{if }}\\gcd(a,m)=1.\\end{cases}}}$
The German mathematician Peter Gustav Lejeune Dirichlet—for whom the character is named—introduced these functions in his 1837 paper on primes in arithmetic progressions.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Dirichlet_character)

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