Über die Anzahl der Primzahlen unter einer gegebenen Grösse
E47355
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.
All labels observed (6)
How this entity was disambiguated
This entity first appeared as the object of triple T373792 — 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: Über die Anzahl der Primzahlen unter einer gegebenen Grösse Context triple: [Bernhard Riemann, notableWork, Über die Anzahl der Primzahlen unter einer gegebenen Grösse]
-
A.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
-
B.
Disquisitiones Arithmeticae
Disquisitiones Arithmeticae is a foundational 1801 treatise on number theory that systematically developed the subject and introduced many of its central concepts and methods.
-
C.
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.
-
D.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
-
E.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic 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: Über die Anzahl der Primzahlen unter einer gegebenen Grösse Target entity description: 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.
-
A.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
-
B.
Disquisitiones Arithmeticae
Disquisitiones Arithmeticae is a foundational 1801 treatise on number theory that systematically developed the subject and introduced many of its central concepts and methods.
-
C.
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.
-
D.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
-
E.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical paper
ⓘ
scientific article ⓘ work on number theory ⓘ |
| alsoKnownAs |
On the Number of Primes Less Than a Given Magnitude
ⓘ
Riemann’s 1859 memoir on primes ⓘ |
| associatedConjecture |
Riemann Hypothesis
ⓘ
linked to:
Riemann hypothesis
|
| author | Bernhard Riemann ⓘ |
| authorBirthYear | 1826 ⓘ |
| authorDeathYear | 1866 ⓘ |
| centuryOfPublication | 19th century ⓘ |
| concerns |
approximation of π(x) by the logarithmic integral li(x)
ⓘ
asymptotic behavior of the prime-counting function π(x) ⓘ error term in the prime number theorem ⓘ |
| countryOfPublication |
Kingdom of Prussia
ⓘ
linked to:
Prussia
|
| defines | Riemann zeta function ⓘ |
| field |
analytic number theory
ⓘ
complex analysis ⓘ number theory ⓘ |
| historicalSignificance |
foundational work of analytic number theory
ⓘ
introduced complex-analytic methods into the study of primes ⓘ |
| influenceOn |
Hilbert–Pólya conjecture
ⓘ
development of analytic number theory ⓘ modern research on L-functions ⓘ prime number theorem ⓘ spectral interpretations of zeros of ζ(s) ⓘ |
| institutionalContext |
Königlich Preußische Akademie der Wissenschaften zu Berlin
ⓘ
linked to:
Prussian Academy of Sciences
|
| introduces |
Riemann Hypothesis
ⓘ
linked to:
Riemann hypothesis
|
| length | approximately 8 pages ⓘ |
| originalLanguage | German ⓘ |
| proposes | all non-trivial zeros of the Riemann zeta function have real part one-half ⓘ |
| publicationDate | 1859-11-19 ⓘ |
| publicationYear | 1859 ⓘ |
| publishedIn | Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin ⓘ |
| states | explicit formula relating π(x) to zeros of ζ(s) ⓘ |
| topic |
Euler product
ⓘ
Riemann Hypothesis ⓘ
linked to:
Riemann hypothesis
Riemann zeta function ⓘ analytic continuation ⓘ distribution of prime numbers ⓘ explicit formulas in prime number theory ⓘ functional equation of the Riemann zeta function ⓘ non-trivial zeros of the zeta function ⓘ prime-counting function ⓘ |
| usesMethod |
analytic continuation of ζ(s)
ⓘ
complex analysis of Dirichlet series ⓘ contour integration ⓘ functional equation of ζ(s) ⓘ |
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: Über die Anzahl der Primzahlen unter einer gegebenen Grösse Description of subject: 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.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
Riemann hypothesis
→
statedIn
→
Riemann's 1859 paper "Über die Anzahl der Primzahlen unter einer gegebenen Grösse"
ⓘ
linked to: Über die Anzahl der Primzahlen unter einer gegebenen Grösse
Über die Anzahl der Primzahlen unter einer gegebenen Grösse
→
alsoKnownAs
→
On the Number of Primes Less Than a Given Magnitude
ⓘ
linked to: Über die Anzahl der Primzahlen unter einer gegebenen Grösse
Über die Anzahl der Primzahlen unter einer gegebenen Grösse
→
alsoKnownAs
→
Riemann’s 1859 memoir on primes
ⓘ
linked to: Über die Anzahl der Primzahlen unter einer gegebenen Grösse
Riemann zeta function
→
historicalPaper
→
Riemann 1859 memoir "On the Number of Primes Less Than a Given Magnitude"
ⓘ
linked to: Über die Anzahl der Primzahlen unter einer gegebenen Grösse
Riemann xi function
→
appearsIn
→
Riemann’s 1859 memoir on the number of primes less than a given magnitude
ⓘ
linked to: Über die Anzahl der Primzahlen unter einer gegebenen Grösse