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.