Dirichlet character

English

Dirichlet character (plural Dirichlet characters)

(analytic number theory) A complex-valued arithmetic function ${\\displaystyle \\chi$ that satisfies the following conditions for some positive integer ${\\displaystyle m}$ and all integers ${\\displaystyle a}$ and ${\\displaystyle b}$ :

1)

{\\displaystyle \\chi (ab)=\\chi (a)\\chi (b);}

2)

{\\displaystyle \\chi (a){\\begin{cases}=0&{\\text{if }}\\;\\gcd(a,m)>1\\\\\eq 0&{\\text{if }}\\;\\gcd(a,m)=1.\\end{cases}}}

(gcd is the greatest common divisor)

3)

{\\displaystyle \\chi (a+m)=\\chi (a)}

.

单词	Dirichlet character
释义	Dirichlet character English Noun Dirichlet character (plural Dirichlet characters) (analytic number theory) A complex-valued arithmetic function ${\\displaystyle \\chi$ that satisfies the following conditions for some positive integer ${\\displaystyle m}$ and all integers ${\\displaystyle a}$ and ${\\displaystyle b}$ : 1) ${\\displaystyle \\chi (ab)=\\chi (a)\\chi (b);}$ 2) ${\\displaystyle \\chi (a){\\begin{cases}=0&{\\text{if }}\\;\\gcd(a,m)>1\\\\\eq 0&{\\text{if }}\\;\\gcd(a,m)=1.\\end{cases}}}$ (gcd is the greatest common divisor) 3) ${\\displaystyle \\chi (a+m)=\\chi (a)}$ .
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