Dirichlet characters

E459563

arithmetic function mathematical concept

Dirichlet characters are completely multiplicative periodic arithmetic functions modulo an integer, fundamental in analytic number theory for constructing Dirichlet L-functions and studying the distribution of primes in arithmetic progressions.

Try in SPARQL Jump to: Surface forms Statements Referenced by

Observed surface forms (3)

Surface form	Occurrences
Dirichlet character	2
Dirichlet character modulo p	1
Dirichlet characters modulo q	1

Statements (47)

Predicate	Object
instanceOf	arithmetic function ⓘ mathematical concept ⓘ
associatedWith	Dirichlet L-functions NERFINISHED ⓘ characters of (Z/nZ)× ⓘ
definedOver	integers modulo n ⓘ
equivalentTo	group homomorphisms from (Z/nZ)× to C× extended by 0 ⓘ
field	analytic number theory ⓘ number theory ⓘ
forms	finite abelian group under pointwise multiplication ⓘ
hasType	imprimitive character ⓘ non-principal character ⓘ primitive character ⓘ principal character ⓘ
mapsFrom	integers ⓘ
mapsTo	complex numbers ⓘ
namedAfter	Peter Gustav Lejeune Dirichlet NERFINISHED ⓘ
orthogonalityProperty	sum over characters χ of χ(a)overline{χ(b)} = φ(N) if a ≡ b mod N ⓘ sum over n mod N of χ(n) = 0 for non-principal χ ⓘ
period	modulus n ⓘ
property	completely multiplicative ⓘ multiplicative arithmetic function ⓘ periodic ⓘ
relatedTo	Dirichlet convolution NERFINISHED ⓘ Möbius function ⓘ Ramanujan sums NERFINISHED ⓘ
satisfies	χ(1) = 1 ⓘ χ(mn) = χ(m)χ(n) ⓘ χ(n + kN) = χ(n) for all integers k ⓘ χ(n) = 0 if gcd(n,N) > 1 ⓘ
usedFor	Gauss sums NERFINISHED ⓘ analytic continuation of L-functions ⓘ character sums ⓘ construction of Dirichlet L-functions ⓘ distribution of residues of primes ⓘ functional equations of L-functions ⓘ non-vanishing results for L-functions ⓘ orthogonality relations in number theory ⓘ proof of Dirichlet’s theorem on arithmetic progressions ⓘ proofs of equidistribution in residue classes ⓘ study of primes in arithmetic progressions ⓘ zero-free regions of L-functions ⓘ
usedIn	Burgess bounds for character sums ⓘ class field theory via Hecke characters ⓘ generalized Riemann hypothesis formulations ⓘ large sieve inequalities ⓘ proofs of the prime number theorem in arithmetic progressions ⓘ
valuesLieIn	roots of unity ⓘ

Referenced by (14)

Full triples — surface form annotated when it differs from this entity's canonical label.

generalized Riemann hypothesis → appliesTo → Dirichlet characters ⓘ

Dirichlet L-functions → basedOn → Dirichlet characters ⓘ

quadratic reciprocity law → formalizedUsing → Dirichlet characters ⓘ

Deuring–Heilbronn phenomenon → involves → Dirichlet characters ⓘ

this entity surface form: Dirichlet character

generalized Riemann hypothesis → involves → Dirichlet characters ⓘ

this entity surface form: Dirichlet characters modulo q

Peter Gustav Lejeune Dirichlet → notableWork → Dirichlet characters ⓘ

Jacobi symbol → relatedConcept → Dirichlet characters ⓘ

this entity surface form: Dirichlet character

Legendre symbol → relatedConcept → Dirichlet characters ⓘ

this entity surface form: Dirichlet character modulo p

Gaussian periods → relatedTo → Dirichlet characters ⓘ

Kronecker–Weber theorem → relatedTo → Dirichlet characters ⓘ

Ramanujan’s sum → relatedTo → Dirichlet characters ⓘ

Selberg class → relatedTo → Dirichlet characters ⓘ

A Course in Arithmetic → subject → Dirichlet characters ⓘ

Multiplicative Number Theory → usesConcept → Dirichlet characters ⓘ