Triple

T2815442
Position Surface form Disambiguated ID Type / Status
Subject Euler product formula for the Riemann zeta function E54270 entity
Predicate dependsOn P100 FINISHED
Object Euler’s method of rearranging absolutely convergent series
Euler’s method of rearranging absolutely convergent series is a technique introduced by Leonhard Euler to systematically reorder and manipulate convergent infinite series in order to derive new identities and product expansions, such as those appearing in analytic number theory.
E300760 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: Euler’s method of rearranging absolutely convergent series | Statement: [Euler product formula for the Riemann zeta function, dependsOn, Euler’s method of rearranging absolutely convergent series]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Euler’s method of rearranging absolutely convergent series
Context triple: [Euler product formula for the Riemann zeta function, dependsOn, Euler’s method of rearranging absolutely convergent series]
  • A. Euler–Maclaurin summation formula
    The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
  • B. Euler’s method for numerical integration
    Euler’s method for numerical integration is a simple first-order numerical procedure used to approximate solutions to ordinary differential equations by stepping forward in small increments.
  • C. Euler’s formula for complex exponentials
    Euler’s formula for complex exponentials is the fundamental identity \(e^{i\theta} = \cos\theta + i\sin\theta\), which links complex exponentials with trigonometric functions and underpins much of complex analysis and engineering mathematics.
  • D. Halley’s method for solving equations
    Halley’s method for solving equations is an iterative numerical algorithm, related to and faster-converging than Newton’s method, used to find approximate roots of equations.
  • E. Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe
    Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe is Bernhard Riemann’s seminal 1854 paper that laid foundational ideas for Fourier series and modern real analysis, including the concept now known as the Riemann integral.
  • 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: Euler’s method of rearranging absolutely convergent series
Triple: [Euler product formula for the Riemann zeta function, dependsOn, Euler’s method of rearranging absolutely convergent series]
Generated description
Euler’s method of rearranging absolutely convergent series is a technique introduced by Leonhard Euler to systematically reorder and manipulate convergent infinite series in order to derive new identities and product expansions, such as those appearing in analytic number theory.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Euler’s method of rearranging absolutely convergent series
Target entity description: Euler’s method of rearranging absolutely convergent series is a technique introduced by Leonhard Euler to systematically reorder and manipulate convergent infinite series in order to derive new identities and product expansions, such as those appearing in analytic number theory.
  • A. Euler–Maclaurin summation formula
    The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
  • B. Euler’s method for numerical integration
    Euler’s method for numerical integration is a simple first-order numerical procedure used to approximate solutions to ordinary differential equations by stepping forward in small increments.
  • C. Euler’s formula for complex exponentials
    Euler’s formula for complex exponentials is the fundamental identity \(e^{i\theta} = \cos\theta + i\sin\theta\), which links complex exponentials with trigonometric functions and underpins much of complex analysis and engineering mathematics.
  • D. Halley’s method for solving equations
    Halley’s method for solving equations is an iterative numerical algorithm, related to and faster-converging than Newton’s method, used to find approximate roots of equations.
  • E. Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe
    Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe is Bernhard Riemann’s seminal 1854 paper that laid foundational ideas for Fourier series and modern real analysis, including the concept now known as the Riemann integral.
  • 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_69ab49de0af08190b3da69683be1e728 completed March 6, 2026, 9:40 p.m.
NER Named-entity recognition batch_69abde4d29488190a32461906dd9ea7e completed March 7, 2026, 8:14 a.m.
NED1 Entity disambiguation (via context triple) batch_69afce9f964081909e422aaf1f026dbb completed March 10, 2026, 7:56 a.m.
NEDg Description generation batch_69afcf12e3a0819098f28d31434a0c5f completed March 10, 2026, 7:58 a.m.
NED2 Entity disambiguation (via description) batch_69afcf9c2d308190b111aa8038c9227a completed March 10, 2026, 8 a.m.
Created at: March 6, 2026, 9:59 p.m.