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 (1)
| Label | Occurrences |
|---|---|
| Siegel–Walfisz theorem canonical | 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.
-
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
NERFINISHED
ⓘ
Carl Ludwig Siegel NERFINISHED ⓘ |
| quantifies | error term in the prime number theorem for arithmetic progressions ⓘ |
| relatedTo |
Bombieri–Vinogradov theorem
NERFINISHED
ⓘ
Dirichlet's theorem on arithmetic progressions NERFINISHED ⓘ Generalized Riemann Hypothesis NERFINISHED ⓘ 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 GENERATED ⓘ |
| usedIn |
applications in additive number theory
ⓘ
proofs about primes in short intervals in arithmetic progressions ⓘ sieve methods ⓘ |
| uses |
Dirichlet L-functions
NERFINISHED
ⓘ
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
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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 (1)
Full triples — surface form annotated when it differs from this entity's canonical label.