Siegel–Walfisz theorem
E871402
The Siegel–Walfisz theorem is a result in analytic number theory that gives strong uniform estimates for the distribution of prime numbers in arithmetic progressions with relatively small moduli.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Siegel–Walfisz theorem canonical | 3 |
| Siegel–Walfisz theorem on average over moduli | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10543866 — 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: Siegel–Walfisz theorem Context triple: [Carl Ludwig Siegel, notableWork, Siegel–Walfisz theorem]
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A.
Bombieri–Vinogradov theorem
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.
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B.
Erdős–Wintner theorem
The Erdős–Wintner theorem is a fundamental result in probabilistic number theory that characterizes when an additive arithmetic function has a limiting distribution.
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C.
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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D.
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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E.
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.
- 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: Siegel–Walfisz theorem Target entity description: The Siegel–Walfisz theorem is a result in analytic number theory that gives strong uniform estimates for the distribution of prime numbers in arithmetic progressions with relatively small moduli.
-
A.
Bombieri–Vinogradov theorem
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.
-
B.
Erdős–Wintner theorem
The Erdős–Wintner theorem is a fundamental result in probabilistic number theory that characterizes when an additive arithmetic function has a limiting distribution.
-
C.
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.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
result in number theory
ⓘ
theorem in analytic number theory ⓘ |
| appearsIn | advanced textbooks on analytic number theory ⓘ |
| appliesTo | reduced residue classes modulo q ⓘ |
| assumes | Generalized Riemann Hypothesis is not required ⓘ |
| concerns |
asymptotic formula for π(x;q,a)
ⓘ
error terms independent of the residue class a ⓘ prime numbers in arithmetic progressions with small moduli ⓘ uniformity over all reduced residue classes a modulo q ⓘ |
| context | distribution of primes in residue classes ⓘ |
| dealsWith |
distribution of prime numbers
ⓘ
primes in arithmetic progressions ⓘ |
| ensures | uniformity in the modulus for primes in arithmetic progressions ⓘ |
| errorTermType | O(x exp(-c√(log x))) for some c > 0, uniformly in q up to (log x)^A ⓘ |
| field | analytic number theory ⓘ |
| gives | strong uniform estimates for primes in arithmetic progressions ⓘ |
| givesBound | error term that is smaller than any fixed power of log x ⓘ |
| hasConsequence | primes are evenly distributed among coprime residue classes for small moduli ⓘ |
| historicalPeriod | 20th century mathematics ⓘ |
| implies | equidistribution of primes among reduced residue classes modulo q for small q ⓘ |
| involves |
Euler's totient function φ(q)
ⓘ
logarithmic functions of x ⓘ |
| isToolFor |
bounding primes in specific congruence classes
ⓘ
results on primes represented by polynomials with congruence conditions ⓘ |
| language | mathematical analysis ⓘ |
| namedAfter |
Arnold Walfisz
ⓘ
Carl Ludwig Siegel ⓘ |
| quantifies | error term in the prime number theorem for arithmetic progressions ⓘ |
| relatedTo |
Bombieri–Vinogradov theorem
ⓘ
Dirichlet's theorem on arithmetic progressions ⓘ Generalized Riemann Hypothesis ⓘ
linked to:
generalized Riemann hypothesis
zero-free region for Dirichlet L-functions ⓘ |
| requires | bounds on character sums ⓘ |
| statesRoughly | π(x;q,a) is x/(φ(q) log x) with a very small uniform error for q ≤ (log x)^A ⓘ |
| strengthens | non-uniform versions of the prime number theorem in arithmetic progressions ⓘ |
| strongerThan | classical prime number theorem for arithmetic progressions for small moduli ⓘ |
| subjectArea | prime number theory ⓘ |
| typeOfResult | uniform distribution theorem ⓘ |
| typicalFormulationInvolves | arbitrary constant A > 0 controlling the size of the modulus ⓘ |
| usedIn |
applications in additive number theory
ⓘ
proofs about primes in short intervals in arithmetic progressions ⓘ sieve methods ⓘ |
| uses |
Dirichlet L-functions
ⓘ
analytic properties of Dirichlet characters ⓘ zero-free regions for L-functions ⓘ |
| validFor | moduli q up to a fixed power of log x ⓘ |
How these facts were elicited
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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: Siegel–Walfisz theorem Description of subject: The Siegel–Walfisz theorem is a result in analytic number theory that gives strong uniform estimates for the distribution of prime numbers in arithmetic progressions with relatively small moduli.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Siegel–Walfisz theorem