In analytic number theory and related branches of mathematics, a complex-valued arithmetic function
, (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.
is a complex primitive n-th root of unity:
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.
If the product of two characters is defined by pointwise multiplication
, and the orthogonality relations:[10] The elements of the finite abelian group
by defining and conversely, a Dirichlet character mod
But this article follows Dirichlet in giving a direct and constructive account of them.
This is partly for historical reasons, in that Dirichlet's work preceded by several decades the development of group theory, and partly for a mathematical reason, namely that the group in question has a simple and interesting structure which is obscured if one treats it as one treats the general Abelian group.
This implies there are only a finite number of characters for a given modulus.
defined by pointwise multiplication: The principal character is an identity: 9) Let
Then The complex conjugate of a root of unity is also its inverse (see here for details), so for
10) The multiplication and identity defined in 8) and the inversion defined in 9) turn the set of Dirichlet characters for a given modulus into a finite abelian group.
; a generator is called a primitive root mod
has a unique factorization as the product of characters mod the prime powers dividing
If the modulus and conductor are equal the character is primitive, otherwise imprimitive.
is defined as its nonzero values are determined by the character mod
The smallest period of the nonzero values is the conductor of the character.
As in the prime-power case, if the conductor equals the modulus the character is primitive, otherwise imprimitive.
If imprimitive it is induced from the character with the smaller modulus.
[36] Primitive characters often simplify (or make possible) formulas in the theories of L-functions[37] and modular forms.
This distinction appears in the functional equation of the Dirichlet L-function.
His later proof, valid for all moduli, was based on his class number formula.
If the modulus is the absolute value of a fundamental discriminant there is a real primitive character (there are two if the modulus is a multiple of 8); otherwise if there are any primitive characters[36] they are imaginary.
Dirichlet characters appear several places in the theory of modular forms and functions.
If define Then See theta series of a Dirichlet character for another example.
It is not necessary to establish the defining properties 1) – 3) to show that a function is a Dirichlet character.
[49] A Dirichlet character is a completely multiplicative function
This equivalent definition of Dirichlet characters was conjectured by Chudakov[51] in 1956, and proved in 2017 by Klurman and Mangerel.