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Concept
Dirichlet character
Summary
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 :
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:
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.
is Euler's totient function.
is a complex primitive n-th root of unity:
but
is the group of units mod . It has order
is the group of Dirichlet characters mod .
etc. are prime numbers.
is a standard abbreviation for
etc. are Dirichlet characters. (the lowercase Greek letter chi for character)
There is no standard notation for Dirichlet characters that includes the modulus. In many contexts (such as in the proof of Dirichlet's theorem) the modulus is fixed. In other contexts, such as this article, characters of different moduli appear. Where appropriate this article employs a variation of Conrey labeling (introduced by Brian Conrey and used by the LMFDB).
In this labeling characters for modulus are denoted where the index is described in the section the group of characters below. In this labeling, denotes an unspecified character and
denotes the principal character mod .
The word "character" is used several ways in mathematics. In this section it refers to a homomorphism from a group (written multiplicatively) to the multiplicative group of the field of complex numbers:
The set of characters is denoted If the product of two characters is defined by pointwise multiplication the identity by the trivial character and the inverse by complex inversion then becomes an abelian group.
Official source
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