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.

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All labels observed (4)

Label	Occurrences
Dirichlet characters canonical	22
Dirichlet character	4
Dirichlet characters modulo q	2
Dirichlet character modulo p	1

How this entity was disambiguated

This entity first appeared as the object of triple T4597214 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Dirichlet characters
Context triple: [Kronecker–Weber theorem, relatedTo, Dirichlet characters]

A. Dirichlet L-functions
Dirichlet L-functions are complex analytic functions built from Dirichlet characters that generalize the Riemann zeta function and play a central role in number theory, particularly in the study of primes in arithmetic progressions.
B. Dedekind zeta functions
Dedekind zeta functions are number-theoretic functions attached to algebraic number fields that encode their arithmetic properties, such as the distribution of prime ideals and class numbers.
C. Chebyshev functions
Chebyshev functions are arithmetic functions in number theory that encode information about the distribution of prime numbers and play a key role in analytic approaches to the prime number theorem.
D. Dirichlet series
A Dirichlet series is an infinite series of the form ∑ aₙ/nˢ, fundamental in analytic number theory for studying arithmetic functions and L-functions.
E. Chebotarev density theorem
The Chebotarev density theorem is a fundamental result in algebraic number theory that generalizes the prime number theorem to describe how often primes in a number field have a given Frobenius conjugacy class in its Galois group.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Dirichlet characters
Target entity description: 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.

A. Dirichlet L-functions
Dirichlet L-functions are complex analytic functions built from Dirichlet characters that generalize the Riemann zeta function and play a central role in number theory, particularly in the study of primes in arithmetic progressions.
B. Dedekind zeta functions
Dedekind zeta functions are number-theoretic functions attached to algebraic number fields that encode their arithmetic properties, such as the distribution of prime ideals and class numbers.
C. Chebyshev functions
Chebyshev functions are arithmetic functions in number theory that encode information about the distribution of prime numbers and play a key role in analytic approaches to the prime number theorem.
D. Dirichlet series
A Dirichlet series is an infinite series of the form ∑ aₙ/nˢ, fundamental in analytic number theory for studying arithmetic functions and L-functions.
E. Chebotarev density theorem
The Chebotarev density theorem is a fundamental result in algebraic number theory that generalizes the prime number theorem to describe how often primes in a number field have a given Frobenius conjugacy class in its Galois group.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	arithmetic function ⓘ mathematical concept ⓘ
associatedWith	Dirichlet L-functions ⓘ 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 ⓘ
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 ⓘ Möbius function ⓘ Ramanujan sums ⓘ linked to: Ramanujan’s sum
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 ⓘ linked to: Gauss sum 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 ⓘ

How these facts were elicited

Referenced by (29)

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

Kronecker–Weber theorem → relatedTo → Dirichlet characters ⓘ

Peter Gustav Lejeune Dirichlet → notableWork → Dirichlet characters ⓘ

Gaussian periods → relatedTo → Dirichlet characters ⓘ

Multiplicative Number Theory → usesConcept → Dirichlet characters ⓘ

Legendre symbol → relatedConcept → Dirichlet character modulo p ⓘ

linked to: Dirichlet characters

quadratic reciprocity law → formalizedUsing → Dirichlet characters ⓘ

Jacobi symbol → relatedConcept → Dirichlet character ⓘ

linked to: Dirichlet characters

Deuring–Heilbronn phenomenon → involves → Dirichlet character ⓘ

linked to: Dirichlet characters

Selberg class → relatedTo → Dirichlet characters ⓘ

A Course in Arithmetic → subject → Dirichlet characters ⓘ

generalized Riemann hypothesis → appliesTo → Dirichlet characters ⓘ

generalized Riemann hypothesis → involves → Dirichlet characters modulo q ⓘ

linked to: Dirichlet characters

Dirichlet L-functions → basedOn → Dirichlet characters ⓘ

Ramanujan’s sum → relatedTo → Dirichlet characters ⓘ

Vinogradov's three-primes theorem → usesConcept → Dirichlet characters ⓘ

Dirichlet → knownFor → Dirichlet characters ⓘ

Dirichlet's theorem on arithmetic progressions → usesConcept → Dirichlet characters ⓘ

Dirichlet's theorem on arithmetic progressions → relatedTo → Dirichlet character ⓘ

linked to: Dirichlet characters

Dirichlet convolution → appliesTo → Dirichlet characters ⓘ

algebraic number theory → fieldOfStudy → Dirichlet characters ⓘ

Bombieri–Vinogradov theorem → involves → Dirichlet characters ⓘ

Vorlesungen über Zahlentheorie → associatedWith → Dirichlet characters ⓘ

subject linked to: Vorlesungen über Zahlentheorie (with Dirichlet)

Heilbronn–Halberstam theorem → usesConcept → Dirichlet characters ⓘ

subject linked to: Heilbronn Halberstam

Halász theorem → relatedTo → Dirichlet characters modulo q ⓘ

linked to: Dirichlet characters

Gauss sum → definitionInvolves → Dirichlet character ⓘ

linked to: Dirichlet characters

Jacobi sums → relatedTo → Dirichlet characters ⓘ

Hecke characters → generalize → Dirichlet characters ⓘ

Stark conjectures → involves → Dirichlet characters ⓘ

GRH → relatedTo → Dirichlet characters ⓘ

All labels observed (4)

How this entity was disambiguated Show

Statements (47)

How these facts were elicited Show

Referenced by (29)

How this entity was disambiguated

How these facts were elicited