Triple

T2566432
Position Surface form Disambiguated ID Type / Status
Subject Johann Heinrich Lambert E57361 entity
Predicate knownFor P22 FINISHED
Object Lambert series
Lambert series are special infinite series in number theory and analysis, often involving arithmetic functions and powers of a variable, introduced by Johann Heinrich Lambert and used in the study of modular forms and q-series.
E279122 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: Lambert series | Statement: [Johann Heinrich Lambert, knownFor, Lambert series]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Lambert series
Context triple: [Johann Heinrich Lambert, knownFor, Lambert series]
  • A. Hahn series
    Hahn series are formal power series with exponents in an ordered abelian group and well-ordered supports, providing a general framework for constructing large ordered fields that include structures like the surreal numbers.
  • B. Bernoulli numbers
    Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
  • C. 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.
  • D. Riemann–Siegel formula
    The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
  • E. Taylor series
    A Taylor series is an infinite sum of terms calculated from the derivatives of a function at a single point, used to represent and approximate functions as power series.
  • 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: Lambert series
Triple: [Johann Heinrich Lambert, knownFor, Lambert series]
Generated description
Lambert series are special infinite series in number theory and analysis, often involving arithmetic functions and powers of a variable, introduced by Johann Heinrich Lambert and used in the study of modular forms and q-series.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Lambert series
Target entity description: Lambert series are special infinite series in number theory and analysis, often involving arithmetic functions and powers of a variable, introduced by Johann Heinrich Lambert and used in the study of modular forms and q-series.
  • A. Hahn series
    Hahn series are formal power series with exponents in an ordered abelian group and well-ordered supports, providing a general framework for constructing large ordered fields that include structures like the surreal numbers.
  • B. Bernoulli numbers
    Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
  • C. 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.
  • D. Riemann–Siegel formula
    The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
  • E. Taylor series
    A Taylor series is an infinite sum of terms calculated from the derivatives of a function at a single point, used to represent and approximate functions as power series.
  • 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_69ab4a4ef9008190a0e6d4422b9418b7 completed March 6, 2026, 9:42 p.m.
NER Named-entity recognition batch_69abd35ef22c8190966612cc75f69eca completed March 7, 2026, 7:27 a.m.
NED1 Entity disambiguation (via context triple) batch_69af6565e05081909dc12aa3240de5f2 completed March 10, 2026, 12:27 a.m.
NEDg Description generation batch_69af667c6b008190b3960f29f5e07653 completed March 10, 2026, 12:31 a.m.
NED2 Entity disambiguation (via description) batch_69af6740cd2c8190a76309238340bd22 completed March 10, 2026, 12:35 a.m.
Created at: March 6, 2026, 9:48 p.m.