Chebyshev’s estimates for π(x)

E898473

bound on the prime-counting function mathematical theorem result in analytic number theory

Chebyshev’s estimates for π(x) are 19th-century bounds on the prime-counting function that showed it grows on the order of x/log x and provided a crucial precursor to the prime number theorem.

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Observed surface forms (1)

Surface form	Occurrences
Chebyshev’s theorem on the distribution of prime numbers	1

Statements (34)

Predicate	Object
instanceOf	bound on the prime-counting function ⓘ mathematical theorem ⓘ result in analytic number theory ⓘ
concerns	asymptotic behavior of π(x) ⓘ distribution of prime numbers ⓘ
era	pre-Riemann prime number theory ⓘ
establishes	nontrivial bounds on π(x) for large x ⓘ
field	analytic number theory ⓘ number theory ⓘ
historicalImportance	first strong evidence for the asymptotic behavior of π(x) ⓘ key step toward the proof of the prime number theorem ⓘ
implies	π(x) = Θ(x / log x) ⓘ
influenceOn	development of analytic methods in number theory ⓘ later proofs of the prime number theorem ⓘ
involves	inequalities for θ(x) ⓘ inequalities for ψ(x) ⓘ
mainSubject	prime-counting function ⓘ
namedAfter	Pafnuty Chebyshev NERFINISHED ⓘ
precursorTo	prime number theorem NERFINISHED ⓘ
provedBy	Pafnuty Chebyshev NERFINISHED ⓘ
provides	lower bounds for π(x) ⓘ upper bounds for π(x) ⓘ
relatedTo	Chebyshev functions NERFINISHED ⓘ Chebyshev’s bias NERFINISHED ⓘ prime number theorem NERFINISHED ⓘ
shows	there exist constants A and B such that A x / log x ≤ π(x) ≤ B x / log x for large x ⓘ
showsGrowthRate	π(x) is of order x / log x ⓘ
symbolOfMainFunction	π(x) ⓘ
timePeriod	19th century ⓘ
type	asymptotic estimate ⓘ inequality ⓘ
uses	Chebyshev functions θ(x) and ψ(x) ⓘ elementary analytic methods ⓘ properties of binomial coefficients ⓘ

Referenced by (2)

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

prime number theorem → predecessorResult → Chebyshev’s estimates for π(x) ⓘ

Pafnuty Chebyshev → notableWork → Chebyshev’s estimates for π(x) ⓘ

this entity surface form: Chebyshev’s theorem on the distribution of prime numbers