Dirichlet characters
E459563
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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Dirichlet characters canonical | 10 |
| Dirichlet character | 2 |
| Dirichlet character modulo p | 1 |
| Dirichlet characters modulo q | 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
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Dirichlet characters Description of subject: 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.
Referenced by (14)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Dirichlet character modulo p
this entity surface form:
Dirichlet character
this entity surface form:
Dirichlet character
this entity surface form:
Dirichlet characters modulo q