Bombieri–Vinogradov theorem
E571012
The Bombieri–Vinogradov theorem is a major result in analytic number theory that gives strong average estimates for the distribution of prime numbers in arithmetic progressions, approaching what is predicted by the Generalized Riemann Hypothesis.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Bombieri–Vinogradov theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6150015 — 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: Bombieri–Vinogradov theorem Context triple: [Enrico Bombieri, knownFor, Bombieri–Vinogradov theorem]
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A.
Dirichlet's theorem on arithmetic progressions
Dirichlet's theorem on arithmetic progressions is a fundamental result in number theory stating that any arithmetic progression with first term and difference coprime contains infinitely many prime numbers.
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B.
Ramanujan–Petersson conjecture
The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
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C.
Bateman–Horn conjecture
The Bateman–Horn conjecture is a far-reaching unproven statement in number theory that predicts how often sets of polynomial expressions simultaneously take prime values, generalizing several earlier conjectures about the distribution of prime numbers.
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D.
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.
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E.
Vinogradov's three-primes theorem
Vinogradov's three-primes theorem is a landmark result in analytic number theory proving that every sufficiently large odd integer can be expressed as the sum of three prime 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: Bombieri–Vinogradov theorem Target entity description: The Bombieri–Vinogradov theorem is a major result in analytic number theory that gives strong average estimates for the distribution of prime numbers in arithmetic progressions, approaching what is predicted by the Generalized Riemann Hypothesis.
-
A.
Dirichlet's theorem on arithmetic progressions
Dirichlet's theorem on arithmetic progressions is a fundamental result in number theory stating that any arithmetic progression with first term and difference coprime contains infinitely many prime numbers.
-
B.
Ramanujan–Petersson conjecture
The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
-
C.
Bateman–Horn conjecture
The Bateman–Horn conjecture is a far-reaching unproven statement in number theory that predicts how often sets of polynomial expressions simultaneously take prime values, generalizing several earlier conjectures about the distribution of prime numbers.
-
D.
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.
-
E.
Vinogradov's three-primes theorem
Vinogradov's three-primes theorem is a landmark result in analytic number theory proving that every sufficiently large odd integer can be expressed as the sum of three prime numbers.
- F. None of above. chosen
Statements (42)
| Predicate | Object |
|---|---|
| instanceOf | theorem in analytic number theory ⓘ |
| appearsIn |
research on distribution of primes in residue classes
ⓘ
research on prime gaps ⓘ |
| appliesTo | moduli q up to about x^{1/2} (with logarithmic savings) ⓘ |
| approximationOf | error term predicted by the Generalized Riemann Hypothesis on average over moduli ⓘ |
| category | results about primes in arithmetic progressions ⓘ |
| comparedWith | Siegel–Walfisz theorem NERFINISHED ⓘ |
| concerns |
distribution of prime numbers in arithmetic progressions
ⓘ
error term in the prime number theorem for arithmetic progressions ⓘ |
| domain | prime numbers ⓘ |
| field | analytic number theory ⓘ |
| generalizationOf | classical average results for primes in arithmetic progressions ⓘ |
| gives | strong average estimates for primes in arithmetic progressions ⓘ |
| hasConsequence |
improved bounds in sieve theory
ⓘ
results on small gaps between primes in arithmetic progressions ⓘ |
| implies | primes are well distributed in arithmetic progressions for most moduli up to about x^{1/2} (up to logarithmic factors) ⓘ |
| improvesOn | Siegel–Walfisz theorem on average over moduli NERFINISHED ⓘ |
| inspired | further work on distribution of primes in arithmetic progressions ⓘ |
| involves |
Dirichlet L-functions
NERFINISHED
ⓘ
Dirichlet characters NERFINISHED ⓘ sums over moduli q ⓘ |
| isToolFor | bounding error terms in arithmetic progression counting functions ⓘ |
| namedAfter |
A. I. Vinogradov
NERFINISHED
ⓘ
Enrico Bombieri NERFINISHED ⓘ |
| provenBy |
A. I. Vinogradov
NERFINISHED
ⓘ
Enrico Bombieri NERFINISHED ⓘ |
| quantifies | average deviation of π(x;q,a) from its expected value x/(φ(q) log x) ⓘ |
| relatedTo |
Dirichlet primes in arithmetic progressions
ⓘ
Elliott–Halberstam conjecture NERFINISHED ⓘ Generalized Riemann Hypothesis NERFINISHED ⓘ |
| requires | zero-free regions for Dirichlet L-functions ⓘ |
| standardReference |
Enrico Bombieri’s paper on the large sieve and its applications to number theory
ⓘ
Iwaniec and Kowalski, Analytic Number Theory NERFINISHED ⓘ |
| status | unconditionally proved theorem ⓘ |
| strengthComparedTo |
comparable to the Generalized Riemann Hypothesis on average over moduli
ⓘ
weaker than the Generalized Riemann Hypothesis pointwise in the modulus ⓘ |
| typeOfResult | average result over moduli ⓘ |
| usedIn |
analytic proofs of results about primes in short intervals
ⓘ
applications to additive problems involving primes ⓘ |
| uses |
large sieve method
ⓘ
zero-density estimates for Dirichlet L-functions ⓘ |
| yearProved | 1965 ⓘ |
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: Bombieri–Vinogradov theorem Description of subject: The Bombieri–Vinogradov theorem is a major result in analytic number theory that gives strong average estimates for the distribution of prime numbers in arithmetic progressions, approaching what is predicted by the Generalized Riemann Hypothesis.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.