Turán–Kubilius inequality
E750083
The Turán–Kubilius inequality is a fundamental result in probabilistic number theory that provides bounds on the distribution of additive arithmetic functions.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Turán–Kubilius inequality canonical | 3 |
| Kubilius model in probabilistic number theory | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T8669825 — 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: Turán–Kubilius inequality Context triple: [Pál Turán, knownFor, Turán–Kubilius inequality]
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A.
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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B.
Khinchin–Kahane type inequalities
Khinchin–Kahane type inequalities are fundamental results in probability and functional analysis that bound moments or norms of random series (often with Rademacher or Gaussian coefficients) in terms of each other, providing powerful tools for studying the geometry of Banach spaces and random processes.
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C.
Erdős–Turán conjecture
The Erdős–Turán conjecture is an unsolved problem in additive number theory asserting that any subset of the positive integers with divergent sum of reciprocals must contain arbitrarily long arithmetic progressions.
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D.
Selberg–Delange method results
Selberg–Delange method results are asymptotic formulas in analytic number theory that precisely describe the average order and distribution of multiplicative arithmetic functions using complex-analytic techniques.
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E.
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.
- 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: Turán–Kubilius inequality Target entity description: The Turán–Kubilius inequality is a fundamental result in probabilistic number theory that provides bounds on the distribution of additive arithmetic functions.
-
A.
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.
-
B.
Khinchin–Kahane type inequalities
Khinchin–Kahane type inequalities are fundamental results in probability and functional analysis that bound moments or norms of random series (often with Rademacher or Gaussian coefficients) in terms of each other, providing powerful tools for studying the geometry of Banach spaces and random processes.
-
C.
Erdős–Turán conjecture
The Erdős–Turán conjecture is an unsolved problem in additive number theory asserting that any subset of the positive integers with divergent sum of reciprocals must contain arbitrarily long arithmetic progressions.
-
D.
Selberg–Delange method results
Selberg–Delange method results are asymptotic formulas in analytic number theory that precisely describe the average order and distribution of multiplicative arithmetic functions using complex-analytic techniques.
-
E.
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.
- F. None of above. chosen
Statements (35)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical inequality
ⓘ
result in probabilistic number theory ⓘ |
| appearsIn |
research on additive functions
ⓘ
texts on probabilistic number theory ⓘ |
| appliesTo |
additive arithmetic function
ⓘ
strongly additive arithmetic function ⓘ |
| assumption | additivity of the arithmetic function ⓘ |
| compares | values of an additive function to its expected value ⓘ |
| context |
metric number theory
ⓘ
probabilistic methods in number theory ⓘ |
| field |
number theory
ⓘ
probabilistic number theory ⓘ |
| hasConsequence | almost all integers have typical additive-function behavior ⓘ |
| historicalPeriod | 20th-century mathematics ⓘ |
| implies | concentration of additive functions around their mean ⓘ |
| mathematicalArea |
analytic number theory
ⓘ
probability theory on arithmetic functions ⓘ |
| namedAfter |
J. Kubilius
ⓘ
Pál Turán ⓘ |
| provides | bounds on the distribution of additive arithmetic functions ⓘ |
| relatedTo |
Erdős–Kac theorem
ⓘ
distribution of prime factors ⓘ normal order of arithmetic functions ⓘ |
| relatesConcept |
distribution of additive functions over integers
ⓘ
mean value of an additive function ⓘ variance of an additive function ⓘ |
| subjectOf | additive arithmetic functions ⓘ |
| typeOfBound | mean-square bound ⓘ |
| typicalDomain | positive integers ⓘ |
| usedFor |
estimating variance of additive arithmetic functions
ⓘ
proving normal order results for arithmetic functions ⓘ studying distribution of values of additive functions ⓘ |
| usedIn |
analysis of additive functions like log n
ⓘ
analysis of additive functions like Ω(n) ⓘ analysis of additive functions like ω(n) ⓘ |
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: Turán–Kubilius inequality Description of subject: The Turán–Kubilius inequality is a fundamental result in probabilistic number theory that provides bounds on the distribution of additive arithmetic functions.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Turán–Kubilius inequality