Erdős–Kac theorem
E554297
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.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
probabilistic number theory result
ⓘ
theorem in number theory ⓘ |
| appliesTo |
additive functions with suitable variance growth
ⓘ
squarefree kernel statistics ⓘ |
| assumption | integers are chosen uniformly from {1,…,x} ⓘ |
| asymptoticDistribution | normal distribution ⓘ |
| classification | result about multiplicative structure of integers ⓘ |
| concernsFunction |
Ω(n)
ⓘ
ω(n) ⓘ |
| describes | distribution of the number of distinct prime factors of integers ⓘ |
| field |
number theory
ⓘ
probabilistic number theory ⓘ |
| hasGeneralizations |
Erdős–Wintner theorem
NERFINISHED
ⓘ
Kubilius model in probabilistic number theory NERFINISHED ⓘ |
| implies | normalized count of prime factors converges in distribution to standard normal ⓘ |
| inspired | further work on probabilistic methods in number theory ⓘ |
| involves |
distribution of additive arithmetic functions
ⓘ
prime factorization of integers ⓘ |
| limitDistribution | standard normal distribution ⓘ |
| limitProcess | x tends to infinity ⓘ |
| mainStatement | the number of distinct prime factors of a typical integer behaves like a normally distributed random variable ⓘ |
| mathematicalArea |
analytic number theory
ⓘ
probability theory ⓘ |
| meanAsymptotic | log log n ⓘ |
| namedAfter |
Mark Kac
NERFINISHED
ⓘ
Paul Erdős NERFINISHED ⓘ |
| normalization | (ω(n) − log log n) / sqrt(log log n) ⓘ |
| normalOrder | ω(n) has normal order log log n ⓘ |
| originalAuthors |
Mark Kac
NERFINISHED
ⓘ
Paul Erdős NERFINISHED ⓘ |
| probabilisticInterpretation | values of arithmetic functions over integers behave like sums of independent random variables ⓘ |
| publicationYear | 1940 ⓘ |
| randomVariableModeled |
Ω(n)
ⓘ
ω(n) ⓘ |
| relatedConcept |
Hardy–Ramanujan theorem
NERFINISHED
ⓘ
Turán–Kubilius inequality NERFINISHED ⓘ central limit theorem ⓘ |
| shows | fluctuations of ω(n) around log log n are typically of size sqrt(log log n) ⓘ |
| strengthens | Hardy–Ramanujan result on normal order of ω(n) ⓘ |
| topic |
additive arithmetic functions
ⓘ
distribution of prime factors ⓘ |
| type |
central limit theorem analogue
ⓘ
limit theorem ⓘ |
| typicalIntegerRange | 1 ≤ n ≤ x with x → ∞ ⓘ |
| usedIn |
analytic number theory
ⓘ
probabilistic models of integers ⓘ |
| varianceAsymptotic | log log n ⓘ |
| yearProved | 1939 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.