Multiplicative Number Theory
E163262
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Multiplicative Number Theory canonical | 1 |
| Multiplicative Number Theory (Harold Davenport) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1428731 — 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: Multiplicative Number Theory Context triple: [Harold Davenport, notableWork, Multiplicative Number Theory]
-
A.
An Introduction to the Theory of Numbers
An Introduction to the Theory of Numbers is a classic textbook in number theory, co-authored by G. H. Hardy, that systematically develops fundamental concepts such as divisibility, prime numbers, Diophantine equations, and quadratic forms.
-
B.
Unsolved Problems in Number Theory
*Unsolved Problems in Number Theory* is a classic reference book that surveys a wide range of open questions and conjectures in number theory, often with historical context and extensive bibliographic notes.
-
C.
Hardy–Littlewood circle method
The Hardy–Littlewood circle method is a powerful analytic number theory technique that uses complex analysis and Fourier series to study additive problems such as Waring’s problem and the Goldbach conjecture.
-
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.
Statistical Independence in Probability, Analysis and Number Theory
"Statistical Independence in Probability, Analysis and Number Theory" is a mathematical monograph by Mark Kac that explores the concept of independence across probability theory, real analysis, and number theory.
- 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: Multiplicative Number Theory Target entity description: 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.
-
A.
An Introduction to the Theory of Numbers
An Introduction to the Theory of Numbers is a classic textbook in number theory, co-authored by G. H. Hardy, that systematically develops fundamental concepts such as divisibility, prime numbers, Diophantine equations, and quadratic forms.
-
B.
Unsolved Problems in Number Theory
*Unsolved Problems in Number Theory* is a classic reference book that surveys a wide range of open questions and conjectures in number theory, often with historical context and extensive bibliographic notes.
-
C.
Hardy–Littlewood circle method
The Hardy–Littlewood circle method is a powerful analytic number theory technique that uses complex analysis and Fourier series to study additive problems such as Waring’s problem and the Goldbach conjecture.
-
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.
Statistical Independence in Probability, Analysis and Number Theory
"Statistical Independence in Probability, Analysis and Number Theory" is a mathematical monograph by Mark Kac that explores the concept of independence across probability theory, real analysis, and number theory.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
branch of analytic number theory
ⓘ
branch of number theory ⓘ |
| concernsSet |
positive integers
ⓘ
prime numbers ⓘ |
| developedFrom | classical analytic number theory ⓘ |
| fieldOfStudy |
Dirichlet series
ⓘ
arithmetic functions ⓘ prime number distribution ⓘ |
| focusesOn |
factorization properties of integers
ⓘ
multiplicative behavior of arithmetic functions ⓘ |
| hasClassicResult |
Dirichlet’s theorem on arithmetic progressions
ⓘ
Erdős–Wintner theorem ⓘ Halász theorem ⓘ Mertens theorems ⓘ
linked to:
Mertens’ theorems
Selberg–Delange method results ⓘ prime number theorem for arithmetic progressions ⓘ |
| hasKeyTool |
Dirichlet generating functions
ⓘ
linked to:
Dirichlet series
Euler product expansions ⓘ
linked to:
Euler products for automorphic L-functions
Mellin transforms ⓘ orthogonality of characters ⓘ |
| hasTextbook |
Multiplicative Number Theory (Harold Davenport)
ⓘ
linked to:
Multiplicative Number Theory
Multiplicative Number Theory I. Classical Theory (Hugh L. Montgomery, Robert C. Vaughan) ⓘ Multiplicative Number Theory II. Analytic and Probabilistic Number Theory (Hugh L. Montgomery, Robert C. Vaughan) ⓘ |
| methodUsed |
Perron’s formula
ⓘ
linked to:
Tauberian theorems
Tauberian theorems ⓘ complex analysis ⓘ contour integration ⓘ sieve methods ⓘ |
| relatedTo |
Dirichlet L-functions
ⓘ
Riemann Hypothesis ⓘ
linked to:
Riemann hypothesis
analytic continuation of Dirichlet series ⓘ prime number theorem ⓘ |
| studiesProperty |
Dirichlet’s theorem on primes in arithmetic progressions
ⓘ
average order of arithmetic functions ⓘ distribution of prime numbers in arithmetic progressions ⓘ mean values of multiplicative functions ⓘ zero-free regions of L-functions ⓘ |
| usesConcept |
Dirichlet characters
ⓘ
Dirichlet convolution ⓘ Euler products ⓘ
linked to:
Euler products for automorphic L-functions
L-functions ⓘ Liouville function ⓘ Möbius function ⓘ Riemann zeta function ⓘ completely multiplicative functions ⓘ divisor functions ⓘ multiplicative functions ⓘ multiplicative structure of integers ⓘ |
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: Multiplicative Number Theory Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Multiplicative Number Theory