Dirichlet character
Complex-valued arithmetic function / From Wikipedia, the free encyclopedia
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In analytic number theory and related branches of mathematics, a complex-valued arithmetic function :\mathbb {Z} \rightarrow \mathbb {C} } is a Dirichlet character of modulus (where is a positive integer) if for all integers and :[1]
- that is, is completely multiplicative.
- (gcd is the greatest common divisor)
- ; that is, is periodic with period .
The simplest possible character, called the principal character, usually denoted , (see Notation below) exists for all moduli:[2]
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.[3][4]