Chebyshev’s estimates for π(x)
E898473
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Chebyshev’s estimates for π(x) canonical | 1 |
| Chebyshev’s theorem on the distribution of prime numbers | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10991355 — 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: Chebyshev’s estimates for π(x) Context triple: [prime number theorem, predecessorResult, Chebyshev’s estimates for π(x)]
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A.
Über die Anzahl der Primzahlen unter einer gegebenen Grösse
Über die Anzahl der Primzahlen unter einer gegebenen Grösse is Bernhard Riemann’s seminal 1859 paper that introduced the Riemann zeta function and laid the foundations of analytic number theory, including the famous Riemann Hypothesis.
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B.
Linnik’s theorem on the least prime in an arithmetic progression
Linnik’s theorem on the least prime in an arithmetic progression is a result in analytic number theory that gives an explicit upper bound, depending only on the modulus, for the size of the smallest prime in any given coprime residue class.
-
C.
Legendre’s conjecture on primes between consecutive squares
Legendre’s conjecture on primes between consecutive squares is an unproven statement in number theory asserting that there is always at least one prime number between any two consecutive perfect squares.
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D.
Siegel–Walfisz theorem
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.
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E.
Turán–Kubilius inequality
The Turán–Kubilius inequality is a fundamental result in probabilistic number theory that provides bounds on the distribution of additive arithmetic functions.
- 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: Chebyshev’s estimates for π(x) Target entity description: 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.
-
A.
Über die Anzahl der Primzahlen unter einer gegebenen Grösse
Über die Anzahl der Primzahlen unter einer gegebenen Grösse is Bernhard Riemann’s seminal 1859 paper that introduced the Riemann zeta function and laid the foundations of analytic number theory, including the famous Riemann Hypothesis.
-
B.
Linnik’s theorem on the least prime in an arithmetic progression
Linnik’s theorem on the least prime in an arithmetic progression is a result in analytic number theory that gives an explicit upper bound, depending only on the modulus, for the size of the smallest prime in any given coprime residue class.
-
C.
Legendre’s conjecture on primes between consecutive squares
Legendre’s conjecture on primes between consecutive squares is an unproven statement in number theory asserting that there is always at least one prime number between any two consecutive perfect squares.
-
D.
Siegel–Walfisz theorem
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.
-
E.
Turán–Kubilius inequality
The Turán–Kubilius inequality is a fundamental result in probabilistic number theory that provides bounds on the distribution of additive arithmetic functions.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Chebyshev’s estimates for π(x) Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Chebyshev’s theorem on the distribution of prime numbers