Triple
T7030756
| Position | Surface form | Disambiguated ID | Type / Status |
|---|---|---|---|
| Subject | Multiplicative Number Theory |
E163262
|
entity |
| Predicate | hasTextbook |
P5478
|
FINISHED |
| Object |
Multiplicative Number Theory I. Classical Theory (Hugh L. Montgomery, Robert C. Vaughan)
Multiplicative Number Theory I. Classical Theory (by Hugh L. Montgomery and Robert C. Vaughan) is a foundational graduate-level textbook that systematically develops the classical theory of multiplicative number theory, including Dirichlet characters, L-functions, and the distribution of prime numbers.
|
E638640
|
NE FINISHED |
How this triple was built (4 steps)
Every LLM step that produced this triple, in pipeline order — named-entity classification, the disambiguation choices (the exact options shown, with the pick highlighted), and the generated description. The batch + timestamp of each is in the Provenance table below.
NER
Named-entity recognition
gpt-5-mini
Instruction
Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).
Input
Phrase: Multiplicative Number Theory I. Classical Theory (Hugh L. Montgomery, Robert C. Vaughan) | Statement: [Multiplicative Number Theory, hasTextbook, Multiplicative Number Theory I. Classical Theory (Hugh L. Montgomery, Robert C. Vaughan)]
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Multiplicative Number Theory I. Classical Theory (Hugh L. Montgomery, Robert C. Vaughan) Context triple: [Multiplicative Number Theory, hasTextbook, Multiplicative Number Theory I. Classical Theory (Hugh L. Montgomery, Robert C. Vaughan)]
-
A.
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.
-
B.
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.
-
C.
Selberg–Delange method results
Selberg–Delange method results are asymptotic formulas in analytic number theory that precisely describe the average order and distribution of multiplicative arithmetic functions using complex-analytic techniques.
-
D.
A. Ivić, The Riemann Zeta-Function
"A. Ivić, The Riemann Zeta-Function" is a comprehensive monograph on the analytic theory of the Riemann zeta function, widely regarded as a standard modern reference in analytic number theory.
-
E.
E. C. Titchmarsh, The Theory of the Riemann Zeta-Function
"E. C. Titchmarsh, The Theory of the Riemann Zeta-Function" is a classic monograph in analytic number theory that provides a comprehensive and authoritative treatment of the Riemann zeta function and related topics.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NEDg
Description generation
gpt-5.1
Instruction
Generate a one-sentence description of the target entity. You are given a context triple in the form (subject, predicate, object), where the object is the target entity. # Instructions Use the triple to infer relevant information about the entity. Describe the entity based on what is most defining, well-known. Avoid repeating the information from the triple, unless really essential. # Response Format Return only the sentence: "Description: [one-sentence description of the target entity]"
Input
Entity: Multiplicative Number Theory I. Classical Theory (Hugh L. Montgomery, Robert C. Vaughan) Triple: [Multiplicative Number Theory, hasTextbook, Multiplicative Number Theory I. Classical Theory (Hugh L. Montgomery, Robert C. Vaughan)]
Generated description
Multiplicative Number Theory I. Classical Theory (by Hugh L. Montgomery and Robert C. Vaughan) is a foundational graduate-level textbook that systematically develops the classical theory of multiplicative number theory, including Dirichlet characters, L-functions, and the distribution of prime numbers.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Multiplicative Number Theory I. Classical Theory (Hugh L. Montgomery, Robert C. Vaughan) Target entity description: Multiplicative Number Theory I. Classical Theory (by Hugh L. Montgomery and Robert C. Vaughan) is a foundational graduate-level textbook that systematically develops the classical theory of multiplicative number theory, including Dirichlet characters, L-functions, and the distribution of prime numbers.
-
A.
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.
-
B.
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.
-
C.
Selberg–Delange method results
Selberg–Delange method results are asymptotic formulas in analytic number theory that precisely describe the average order and distribution of multiplicative arithmetic functions using complex-analytic techniques.
-
D.
A. Ivić, The Riemann Zeta-Function
"A. Ivić, The Riemann Zeta-Function" is a comprehensive monograph on the analytic theory of the Riemann zeta function, widely regarded as a standard modern reference in analytic number theory.
-
E.
E. C. Titchmarsh, The Theory of the Riemann Zeta-Function
"E. C. Titchmarsh, The Theory of the Riemann Zeta-Function" is a classic monograph in analytic number theory that provides a comprehensive and authoritative treatment of the Riemann zeta function and related topics.
- F. None of above. chosen
Provenance (5 batches)
The batch behind each pipeline step, in order, with when it ran. Timestamps are batch-level — stages were processed in waves, so the object chain (NER → NED1 → NEDg → NED2) reads in order, but predicate / elicitation batches can sit in a different wave.
| Step | Stage | Batch ID | Status | When |
|---|---|---|---|---|
| creating | Elicitation | batch_69c6885d691c81908cf7d31083113886 |
completed | March 27, 2026, 1:38 p.m. |
| NER | Named-entity recognition | batch_69c6e77415e88190ab65137382f1b155 |
completed | March 27, 2026, 8:24 p.m. |
| NED1 | Entity disambiguation (via context triple) | batch_69c7885d83d4819099cc334dd2841f3b |
completed | March 28, 2026, 7:50 a.m. |
| NEDg | Description generation | batch_69c789c962b081909cff8b58c87f224e |
completed | March 28, 2026, 7:56 a.m. |
| NED2 | Entity disambiguation (via description) | batch_69c78a9675b4819087836dfc438df9f0 |
completed | March 28, 2026, 8 a.m. |
Created at: March 27, 2026, 2:35 p.m.