In analytic number theory and related branches of mathematics, a complex-valued arithmetic function 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]
^This is the standard definition; e.g. Davenport p.27; Landau p. 109; Ireland and Rosen p. 253
^Note the special case of modulus 1: the unique character mod 1 is the constant 1; all other characters are 0 at 0
^Davenport p. 1
^An English translation is in External Links
and 24 Related for: Dirichlet character information
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