Dirichlet series
E300759
A Dirichlet series is an infinite series of the form ∑ aₙ/nˢ, fundamental in analytic number theory for studying arithmetic functions and L-functions.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Dirichlet series canonical | 6 |
| Dirichlet generating function | 1 |
| Dirichlet generating functions | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2815417 — 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 series Context triple: [Euler product formula for the Riemann zeta function, relatesTo, Dirichlet series]
-
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.
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B.
Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
-
C.
Lambert series
Lambert series are special infinite series in number theory and analysis, often involving arithmetic functions and powers of a variable, introduced by Johann Heinrich Lambert and used in the study of modular forms and q-series.
-
D.
Selberg class
The Selberg class is a collection of Dirichlet series with specific analytic properties introduced to generalize and axiomatize L-functions in number theory.
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E.
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.
- 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 series Target entity description: A Dirichlet series is an infinite series of the form ∑ aₙ/nˢ, fundamental in analytic number theory for studying arithmetic functions and L-functions.
-
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.
Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
-
C.
Lambert series
Lambert series are special infinite series in number theory and analysis, often involving arithmetic functions and powers of a variable, introduced by Johann Heinrich Lambert and used in the study of modular forms and q-series.
-
D.
Selberg class
The Selberg class is a collection of Dirichlet series with specific analytic properties introduced to generalize and axiomatize L-functions in number theory.
-
E.
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.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
analytic number theory concept
ⓘ
complex function ⓘ mathematical series ⓘ |
| canHaveEulerProduct | when coefficients are multiplicative ⓘ |
| coefficientSequence | (a_n)_{n≥1} ⓘ |
| convergesIn | right half-plane Re(s) > σ_c ⓘ |
| domainOfDefinition | subset of the complex plane ⓘ |
| encodes | information about coefficients a_n ⓘ |
| field |
analytic number theory
ⓘ
number theory ⓘ |
| generalizationOf | ordinary generating function with n^{-s} weights ⓘ |
| hasAbscissaOfAbsoluteConvergence | σ_a ⓘ |
| hasAbscissaOfConvergence | σ_c ⓘ |
| hasAbscissaOfUniformConvergence | σ_u ⓘ |
| hasGeneralForm | ∑_{n=1}^{∞} a_n n^{-s} ⓘ |
| hasProperty |
absolute convergence implies uniform convergence on compact subsets of half-planes
ⓘ
uniqueness of coefficients in domain of convergence ⓘ |
| hasTypicalRegionOfAnalyticity | half-plane Re(s) > σ_c ⓘ |
| isStudiedIn |
complex analysis
ⓘ
harmonic analysis ⓘ probabilistic number theory ⓘ |
| isToolFor |
Tauberian theorems
ⓘ
analytic continuation ⓘ functional equations of L-functions ⓘ prime number theorems in arithmetic progressions ⓘ |
| isUsedToStudy |
L-functions
ⓘ
arithmetic functions ⓘ distribution of prime numbers ⓘ multiplicative functions ⓘ |
| mayAdmit | meromorphic continuation beyond initial half-plane ⓘ |
| namedAfter | Peter Gustav Lejeune Dirichlet ⓘ |
| productCorrespondsTo | Dirichlet convolution of coefficient sequences ⓘ |
| relatedTo |
Euler product
ⓘ
Mellin transforms ⓘ
surface form:
Mellin transform
power series ⓘ |
| satisfiesInequality | σ_c ≤ σ_u ≤ σ_a ⓘ |
| specialCase |
Dirichlet L-functions
ⓘ
surface form:
Dirichlet L-function
Dirichlet series self-linksurface differs ⓘ
surface form:
Dirichlet generating function
Hurwitz zeta function ⓘ Riemann zeta function ⓘ |
| supportsOperation |
Dirichlet convolution via product
ⓘ
termwise addition ⓘ |
| usedIn |
automorphic forms
ⓘ
proofs of Dirichlet’s theorem on arithmetic progressions ⓘ spectral theory of automorphic Laplacians ⓘ study of modular forms ⓘ |
| variable | complex variable s ⓘ |
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 series Description of subject: A Dirichlet series is an infinite series of the form ∑ aₙ/nˢ, fundamental in analytic number theory for studying arithmetic functions and L-functions.
Referenced by (8)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Dirichlet generating functions
this entity surface form:
Dirichlet generating function