Triple

T22964951
Position Surface form Disambiguated ID Type / Status
Subject Bombieri–Vinogradov theorem E571012 entity
Predicate standardReference P33736 FINISHED
Object Iwaniec and Kowalski, Analytic Number Theory
"Iwaniec and Kowalski, Analytic Number Theory" is a comprehensive graduate-level textbook that systematically develops modern analytic methods in number theory, including prime distribution, sieve methods, and L-functions.
E1563174 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: Iwaniec and Kowalski, Analytic Number Theory | Statement: [Bombieri–Vinogradov theorem, standardReference, Iwaniec and Kowalski, Analytic Number Theory]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Iwaniec and Kowalski, Analytic Number Theory
Context triple: [Bombieri–Vinogradov theorem, standardReference, Iwaniec and Kowalski, Analytic Number Theory]
  • A. 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.
  • B. 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.
  • C. 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.
  • D. 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.
  • E. Linnik’s theorem on the least prime in an arithmetic progression
    Linnik’s theorem on the least prime in an arithmetic progression is a result in analytic number theory that gives an explicit upper bound, depending only on the modulus, for the size of the smallest prime in any given coprime residue class.
  • 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: Iwaniec and Kowalski, Analytic Number Theory
Triple: [Bombieri–Vinogradov theorem, standardReference, Iwaniec and Kowalski, Analytic Number Theory]
Generated description
"Iwaniec and Kowalski, Analytic Number Theory" is a comprehensive graduate-level textbook that systematically develops modern analytic methods in number theory, including prime distribution, sieve methods, and L-functions.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Iwaniec and Kowalski, Analytic Number Theory
Target entity description: "Iwaniec and Kowalski, Analytic Number Theory" is a comprehensive graduate-level textbook that systematically develops modern analytic methods in number theory, including prime distribution, sieve methods, and L-functions.
  • A. 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.
  • B. 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.
  • C. 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.
  • D. 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.
  • E. Linnik’s theorem on the least prime in an arithmetic progression
    Linnik’s theorem on the least prime in an arithmetic progression is a result in analytic number theory that gives an explicit upper bound, depending only on the modulus, for the size of the smallest prime in any given coprime residue class.
  • 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_69e245b212a88190b5259caf51606084 completed April 17, 2026, 2:37 p.m.
NER Named-entity recognition batch_69f181f763688190aab8f444a1a71577 completed April 29, 2026, 3:58 a.m.
NED1 Entity disambiguation (via context triple) batch_6a0bca22caa88190889e965c78477a96 completed May 19, 2026, 2:25 a.m.
NEDg Description generation batch_6a0bcb1904c88190ac32e5709bc95858 completed May 19, 2026, 2:29 a.m.
NED2 Entity disambiguation (via description) batch_6a0bcbb8793881909621b16c9ad51cbb completed May 19, 2026, 2:32 a.m.
Created at: April 17, 2026, 3:47 p.m.