Mertens’ theorems
E300762
Mertens’ theorems are classical results in analytic number theory that give precise asymptotic estimates for sums involving the Möbius function and the reciprocals of primes, illuminating the distribution of primes and their connection to the Riemann zeta function.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Mertens theorems | 3 |
| Mertens’ theorems canonical | 2 |
| Mertens’ first theorem | 1 |
| Mertens’ second theorem | 1 |
| Mertens’ third theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2815453 — 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: Mertens’ theorems Context triple: [Euler product formula for the Riemann zeta function, relatedConcept, Mertens’ theorems]
-
A.
Hardy–Ramanujan asymptotic formula
The Hardy–Ramanujan asymptotic formula is a landmark result in number theory that gives an approximate expression for the partition function p(n), describing how the number of integer partitions of n grows rapidly with n.
-
B.
Deuring–Heilbronn phenomenon
The Deuring–Heilbronn phenomenon is a result in analytic number theory describing how the presence of an exceptional (Siegel) zero of a Dirichlet L-function forces other zeros away from the real axis, sharpening zero-free regions and affecting the distribution of primes in arithmetic progressions.
-
C.
Multiplicative Number Theory
Multiplicative Number Theory is a branch of analytic number theory that studies arithmetic functions and prime number distributions through their multiplicative properties and associated Dirichlet series.
-
D.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
-
E.
Lindemann–Weierstrass theorem precursor
The Lindemann–Weierstrass theorem precursor is an early foundational result in transcendental number theory developed by Ferdinand von Lindemann that paved the way for the full Lindemann–Weierstrass theorem on the algebraic independence of exponentials of algebraic numbers.
- 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: Mertens’ theorems Target entity description: Mertens’ theorems are classical results in analytic number theory that give precise asymptotic estimates for sums involving the Möbius function and the reciprocals of primes, illuminating the distribution of primes and their connection to the Riemann zeta function.
-
A.
Hardy–Ramanujan asymptotic formula
The Hardy–Ramanujan asymptotic formula is a landmark result in number theory that gives an approximate expression for the partition function p(n), describing how the number of integer partitions of n grows rapidly with n.
-
B.
Deuring–Heilbronn phenomenon
The Deuring–Heilbronn phenomenon is a result in analytic number theory describing how the presence of an exceptional (Siegel) zero of a Dirichlet L-function forces other zeros away from the real axis, sharpening zero-free regions and affecting the distribution of primes in arithmetic progressions.
-
C.
Multiplicative Number Theory
Multiplicative Number Theory is a branch of analytic number theory that studies arithmetic functions and prime number distributions through their multiplicative properties and associated Dirichlet series.
-
D.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
-
E.
Lindemann–Weierstrass theorem precursor
The Lindemann–Weierstrass theorem precursor is an early foundational result in transcendental number theory developed by Ferdinand von Lindemann that paved the way for the full Lindemann–Weierstrass theorem on the algebraic independence of exponentials of algebraic numbers.
- F. None of above. chosen
Statements (42)
| Predicate | Object |
|---|---|
| instanceOf |
result in number theory
ⓘ
theorem in analytic number theory ⓘ theorem in analytic number theory ⓘ theorem in analytic number theory ⓘ theorem in analytic number theory ⓘ |
| appliesTo |
multiplicative arithmetic functions
ⓘ
prime numbers ⓘ |
| clarifies | connection between primes and the Riemann zeta function ⓘ |
| describes |
asymptotic behavior of the product over primes (1 − 1/p)
ⓘ
asymptotic behavior of the sum of reciprocals of primes ⓘ |
| field |
analytic number theory
ⓘ
number theory ⓘ |
| gives |
precise asymptotic estimates for products over primes
ⓘ
precise asymptotic estimates for sums involving primes ⓘ |
| hasConsequence |
estimates for partial Euler products
ⓘ
information about density of primes ⓘ refined bounds for sums over primes ⓘ |
| hasPart |
Mertens’ first theorem
ⓘ
linked to:
Mertens’ theorems
Mertens’ second theorem ⓘ
linked to:
Mertens’ theorems
Mertens’ third theorem ⓘ
linked to:
Mertens’ theorems
|
| historicalPeriod | 19th century mathematics ⓘ |
| involves |
Euler–Mascheroni constant
ⓘ
linked to:
Euler–Mascheroni constant γ
Möbius function ⓘ iterated logarithm ⓘ natural logarithm ⓘ prime numbers ⓘ sums over primes ⓘ |
| mainTopic |
Möbius function
ⓘ
Riemann zeta function ⓘ distribution of prime numbers ⓘ |
| namedAfter |
Austrian mathematician Franz Mertens
ⓘ
Franz Mertens ⓘ |
| relatedTo |
Chebyshev’s theorems
ⓘ
linked to:
Chebyshev inequalities
Dirichlet series ⓘ Mertens function ⓘ Prime Number Theorem ⓘ
linked to:
prime number theorem
Riemann Hypothesis ⓘ
linked to:
Riemann hypothesis
|
| statement |
The product_{p \le x} (1 − 1/p) ~ e^{−γ}/log x as x → ∞
ⓘ
The sum_{p \le x} (log p)/p = log x + O(1) as x → ∞ ⓘ The sum_{p \le x} 1/p = log log x + B + o(1) as x → ∞ for a constant B ⓘ |
| usesTool |
complex analysis
ⓘ
properties of the Riemann zeta function ⓘ |
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: Mertens’ theorems Description of subject: Mertens’ theorems are classical results in analytic number theory that give precise asymptotic estimates for sums involving the Möbius function and the reciprocals of primes, illuminating the distribution of primes and their connection to the Riemann zeta function.
Referenced by (8)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Mertens’ theorems
linked to: Mertens’ theorems
linked to: Mertens’ theorems
linked to: Mertens’ theorems
linked to: Mertens’ theorems