Selberg sieve
E246699
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Selberg sieve canonical | 2 |
| Selberg lower bound sieve | 1 |
| Selberg upper bound sieve | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2252105 — 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: Selberg sieve Context triple: [Atle Selberg, knownFor, Selberg sieve]
-
A.
Hardy–Littlewood circle method
The Hardy–Littlewood circle method is a powerful analytic number theory technique that uses complex analysis and Fourier series to study additive problems such as Waring’s problem and the Goldbach conjecture.
-
B.
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.
-
C.
Hardy–Littlewood conjectures
The Hardy–Littlewood conjectures are a collection of influential unproven hypotheses in analytic number theory that generalize the prime number theorem to describe the distribution of prime numbers and prime constellations.
-
D.
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.
-
E.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
- 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: Selberg sieve Target entity description: 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.
-
A.
Hardy–Littlewood circle method
The Hardy–Littlewood circle method is a powerful analytic number theory technique that uses complex analysis and Fourier series to study additive problems such as Waring’s problem and the Goldbach conjecture.
-
B.
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.
-
C.
Hardy–Littlewood conjectures
The Hardy–Littlewood conjectures are a collection of influential unproven hypotheses in analytic number theory that generalize the prime number theorem to describe the distribution of prime numbers and prime constellations.
-
D.
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.
-
E.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
- F. None of above. chosen
Statements (42)
| Predicate | Object |
|---|---|
| instanceOf |
analytic number theory method
ⓘ
mathematical method ⓘ sieve method ⓘ |
| appearsIn |
analytic number theory monographs
ⓘ
sieve theory textbooks ⓘ |
| appliesTo |
distribution of primes in arithmetic progressions
ⓘ
problems about almost primes ⓘ sets of integers with local congruence conditions ⓘ |
| basedOn |
quadratic forms in sieve weights
ⓘ
weight functions ⓘ |
| canGive | lower bounds in certain variants ⓘ |
| characterizedBy |
flexible choice of weight functions
ⓘ
quadratic optimization problem for weights ⓘ |
| contrastedWith | Brun combinatorial sieve ⓘ |
| developedBy | Atle Selberg ⓘ |
| developmentPeriod | 20th century ⓘ |
| field |
analytic number theory
ⓘ
number theory ⓘ |
| gives | upper bounds for sifted sets ⓘ |
| goal | approximate characteristic function of sifted set ⓘ |
| hasVariant |
Selberg sieve
self-linksurface differs
ⓘ
surface form:
Selberg lower bound sieve
Selberg sieve self-linksurface differs ⓘ
surface form:
Selberg upper bound sieve
|
| influenced |
modern sieve theory
ⓘ
research on almost primes ⓘ research on primes in short intervals ⓘ |
| namedAfter | Atle Selberg ⓘ |
| optimizedBy | choice of sieve weights ⓘ |
| relatedTo |
Brun sieve
ⓘ
combinatorial sieve ⓘ large sieve ⓘ sieve of Eratosthenes ⓘ |
| toolFor |
bounding error terms in prime-counting problems
ⓘ
problems on primes represented by polynomials ⓘ |
| typicalAssumption | multiplicative structure of local densities ⓘ |
| usedFor |
bounding the number of integers free of small prime factors
ⓘ
estimating size of sifted sets of integers ⓘ studying distribution of prime numbers ⓘ upper bounds in sieve theory problems ⓘ |
| usesConcept |
Möbius function
ⓘ
divisibility conditions ⓘ inclusion–exclusion principle ⓘ multiplicative 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: Selberg sieve Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Selberg upper bound sieve
this entity surface form:
Selberg lower bound sieve