Dirichlet density
E790518
Dirichlet density is a notion of density for subsets of prime numbers defined via Dirichlet series, used to measure how frequently such primes occur in analytic number theory.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Dirichlet density canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T9297041 — 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 density Context triple: [Chebotarev density theorem, usesConcept, Dirichlet density]
-
A.
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.
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B.
Erdős–Kac theorem
The Erdős–Kac theorem is a fundamental result in probabilistic number theory stating that the number of distinct prime factors of a typical integer behaves like a normally distributed random variable.
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C.
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.
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D.
Deuring–Heilbronn phenomenon
The Deuring–Heilbronn phenomenon is a result in analytic number theory describing how the presence of an exceptional (Siegel) zero of a Dirichlet L-function forces other zeros away from the real axis, sharpening zero-free regions and affecting the distribution of primes in arithmetic progressions.
-
E.
Dirichlet hyperbola method
The Dirichlet hyperbola method is a technique in analytic number theory used to estimate sums of arithmetic functions by splitting double sums along a hyperbola to obtain asymptotic formulas.
- 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 density Target entity description: Dirichlet density is a notion of density for subsets of prime numbers defined via Dirichlet series, used to measure how frequently such primes occur in analytic number theory.
-
A.
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.
-
B.
Erdős–Kac theorem
The Erdős–Kac theorem is a fundamental result in probabilistic number theory stating that the number of distinct prime factors of a typical integer behaves like a normally distributed random variable.
-
C.
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.
-
D.
Deuring–Heilbronn phenomenon
The Deuring–Heilbronn phenomenon is a result in analytic number theory describing how the presence of an exceptional (Siegel) zero of a Dirichlet L-function forces other zeros away from the real axis, sharpening zero-free regions and affecting the distribution of primes in arithmetic progressions.
-
E.
Dirichlet hyperbola method
The Dirichlet hyperbola method is a technique in analytic number theory used to estimate sums of arithmetic functions by splitting double sums along a hyperbola to obtain asymptotic formulas.
- F. None of above. chosen
Statements (30)
| Predicate | Object |
|---|---|
| instanceOf |
concept in analytic number theory
ⓘ
density for subsets of prime numbers ⓘ notion of density ⓘ |
| appliesTo |
Chebotarev sets of primes
NERFINISHED
ⓘ
sets of primes defined by congruence conditions ⓘ |
| basedOn | Dirichlet series NERFINISHED ⓘ |
| comparedWith |
analytic density
ⓘ
natural density ⓘ |
| contrastsWith | asymptotic density defined by counting function π_A(x) ⓘ |
| definedOn | subsets of prime numbers ⓘ |
| dependsOn | behavior of Dirichlet series near s = 1 ⓘ |
| domain | set of prime numbers ⓘ |
| field | analytic number theory ⓘ |
| generalizationOf | natural density for many arithmetic sets of primes ⓘ |
| hasAlternativeName | Dirichlet analytic density NERFINISHED ⓘ |
| hasKeyIdea | encode a set of primes in a Dirichlet series and study its singularity at s = 1 ⓘ |
| hasProperty |
defined via limiting behavior of Dirichlet series
ⓘ
invariant under finite modification of a set of primes ⓘ may exist when natural density does not ⓘ |
| mathematicalObjectType | real number between 0 and 1 for many natural sets of primes ⓘ |
| namedAfter | Johann Peter Gustav Lejeune Dirichlet NERFINISHED ⓘ |
| relatedTo |
Chebotarev density theorem
NERFINISHED
ⓘ
Dirichlet L-functions NERFINISHED ⓘ analytic continuation of Dirichlet series ⓘ prime number theorem for arithmetic progressions ⓘ |
| usedFor |
measuring frequency of subsets of prime numbers
ⓘ
measuring naturalness of sets of primes ⓘ studying distribution of primes in arithmetic progressions ⓘ |
| usedIn |
formulation of density theorems in algebraic number theory
ⓘ
proofs and statements about distribution of primes in residue classes ⓘ |
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 density Description of subject: Dirichlet density is a notion of density for subsets of prime numbers defined via Dirichlet series, used to measure how frequently such primes occur in analytic number theory.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.