analytic number theory
E530316
Analytic number theory is a branch of mathematics that uses tools from mathematical analysis to study the distribution and properties of integers, especially prime numbers.
All labels observed (1)
| Label | Occurrences |
|---|---|
| analytic number theory canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5570638 — 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: analytic number theory Context triple: [Fermat's theorem on sums of two squares, usedIn, analytic number theory]
-
A.
Multiplicative Number Theory
Multiplicative Number Theory is a branch of analytic number theory that studies arithmetic functions and prime number distributions through their multiplicative properties and associated Dirichlet series.
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B.
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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C.
Selberg sieve
The Selberg sieve is a powerful analytic number theory method developed by Atle Selberg for estimating the size of sets of integers filtered by divisibility conditions, particularly in the study of prime numbers.
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D.
L-functions
L-functions are complex analytic functions, often arising from number theory and algebraic geometry, that encode deep arithmetic information and generalize the Riemann zeta function.
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E.
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.
- 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: analytic number theory Target entity description: Analytic number theory is a branch of mathematics that uses tools from mathematical analysis to study the distribution and properties of integers, especially prime numbers.
-
A.
Multiplicative Number Theory
Multiplicative Number Theory is a branch of analytic number theory that studies arithmetic functions and prime number distributions through their multiplicative properties and associated Dirichlet series.
-
B.
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.
-
C.
Selberg sieve
The Selberg sieve is a powerful analytic number theory method developed by Atle Selberg for estimating the size of sets of integers filtered by divisibility conditions, particularly in the study of prime numbers.
-
D.
L-functions
L-functions are complex analytic functions, often arising from number theory and algebraic geometry, that encode deep arithmetic information and generalize the Riemann zeta function.
-
E.
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.
- F. None of above. chosen
Statements (58)
| Predicate | Object |
|---|---|
| instanceOf |
branch of mathematics
ⓘ
subfield of number theory ⓘ |
| appliesTo |
Diophantine equations
ⓘ
arithmetic geometry ⓘ problems about prime distribution ⓘ |
| centralConjecture | Riemann hypothesis ⓘ |
| centralObject | Riemann zeta function ⓘ |
| centralTheorem |
Bombieri–Vinogradov theorem
ⓘ
Chebotarev density theorem ⓘ Dirichlet's theorem on arithmetic progressions ⓘ Pólya–Vinogradov inequality ⓘ Siegel–Walfisz theorem ⓘ Weyl's equidistribution theorem ⓘ
linked to:
theory of uniform distribution modulo 1
prime number theorem ⓘ zero-free regions for L-functions ⓘ |
| focusesOn |
additive properties of integers
ⓘ
distribution of integers in arithmetic progressions ⓘ distribution of prime numbers ⓘ multiplicative properties of integers ⓘ |
| hasSubarea |
additive number theory (analytic methods)
ⓘ
analytic theory of L-functions ⓘ multiplicative number theory ⓘ sieve theory ⓘ |
| historicalFigure |
Atle Selberg
ⓘ
Bernhard Riemann ⓘ Charles-Jean de la Vallée Poussin ⓘ G. H. Hardy ⓘ Harald Cramér ⓘ Ivan Vinogradov ⓘ Jacques Hadamard ⓘ John Edensor Littlewood ⓘ |
| relatedField |
additive combinatorics
ⓘ
algebraic number theory ⓘ probabilistic number theory ⓘ |
| studies |
Goldbach-type problems
ⓘ
Waring-type problems ⓘ additive problems in number theory ⓘ automorphic forms ⓘ distribution of primes in arithmetic progressions ⓘ distribution of primes in short intervals ⓘ distribution of values of arithmetic functions ⓘ integers ⓘ modular forms ⓘ prime gaps ⓘ prime numbers ⓘ zeros of L-functions ⓘ zeros of the Riemann zeta function ⓘ |
| usesTool |
Dirichlet series
ⓘ
Fourier analysis ⓘ L-functions ⓘ Tauberian theorems ⓘ complex analysis ⓘ generating functions ⓘ harmonic analysis ⓘ mathematical analysis ⓘ probabilistic methods ⓘ sieve methods ⓘ zeta functions ⓘ |
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: analytic number theory Description of subject: Analytic number theory is a branch of mathematics that uses tools from mathematical analysis to study the distribution and properties of integers, especially prime numbers.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.