prime number theorem

E259759

mathematical theorem result in analytic number theory

The prime number theorem is a fundamental result in number theory that describes how prime numbers become less frequent and provides an approximate formula for the number of primes less than a given large number.

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All labels observed (2)

Label	Occurrences
prime number theorem canonical	6
Prime Number Theorem	1

Statements (48)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in analytic number theory ⓘ
approximation	π(x) ≈ li(x) ⓘ
approximationQuality	improves as x tends to infinity ⓘ
concernsSet	set of prime numbers ⓘ
describes	asymptotic distribution of prime numbers ⓘ
domain	positive real numbers x ⓘ
elementaryProofBy	Atle Selberg ⓘ Pál Erdős ⓘ surface form: Paul Erdős
elementaryProofYear	1949 ⓘ
equivalentTo	ψ(x) ~ x ⓘ
field	analytic number theory ⓘ number theory ⓘ
firstProofBy	Charles-Jean de la Vallée Poussin ⓘ Jacques Hadamard ⓘ
generalizedTo	Chebotarev density theorem ⓘ prime number theorem for arithmetic progressions ⓘ
hasConsequence	average gap between consecutive primes near x is about log x ⓘ proportion of numbers up to x that are prime is about 1 / log x ⓘ
hasElementaryProof	yes ⓘ
historicalConjectureBy	Adrien-Marie Legendre ⓘ Carl Friedrich Gauss ⓘ
implies	density of primes near x is about 1 / log x ⓘ
involvesFunction	Chebyshev functions ⓘ surface form: Chebyshev function ψ(x) natural logarithm log x ⓘ prime-counting function π(x) ⓘ
language	mathematical notation ⓘ
namedAfter	prime numbers ⓘ
predecessorResult	Chebyshev’s estimates for π(x) ⓘ
provedIndependentlyBy	Charles-Jean de la Vallée Poussin ⓘ Jacques Hadamard ⓘ
publicationYear	1896 ⓘ
refinedBy	error term estimates for π(x) ⓘ logarithmic integral li(x) ⓘ
relatedConcept	Dirichlet's theorem on arithmetic progressions ⓘ surface form: Dirichlet’s theorem on arithmetic progressions Mertens’ theorems ⓘ distribution of primes in short intervals ⓘ prime gaps ⓘ
relatedTo	Riemann hypothesis ⓘ Riemann zeta function ⓘ
states	the number of primes less than x is asymptotic to x / log x ⓘ
symbolicForm	π(x) ~ x / log x ⓘ
topicOf	many advanced textbooks in analytic number theory ⓘ
type	asymptotic formula ⓘ
usesTool	complex analysis ⓘ non-vanishing of the Riemann zeta function on the line Re(s) = 1 ⓘ properties of the Riemann zeta function ⓘ
yearProved	1896 ⓘ

Referenced by (7)

Full triples — surface form annotated when it differs from this entity's canonical label.

Riemann hypothesis → relatedTo → prime number theorem ⓘ

Über die Anzahl der Primzahlen unter einer gegebenen Grösse → influenceOn → prime number theorem ⓘ

Riemann zeta function → connectedTo → prime number theorem ⓘ

Jacques Hadamard → knownFor → prime number theorem ⓘ

Chebotarev density theorem → generalizes → prime number theorem ⓘ

Chebyshev functions → usedToProve → prime number theorem ⓘ

subject surface form: Chebyshev function ψ(x)

Mertens’ theorems → relatedTo → prime number theorem ⓘ

this entity surface form: Prime Number Theorem