Triple

T8644722
Position Surface form Disambiguated ID Type / Status
Subject Deuring–Heilbronn phenomenon E204744 entity
Predicate relatedTo P37 FINISHED
Object Siegel’s theorem on zeros of L-functions
Siegel’s theorem on zeros of L-functions is a result in analytic number theory that gives strong bounds on how close nontrivial zeros of Dirichlet L-functions can approach 1, with deep implications for the distribution of primes in arithmetic progressions.
E747888 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: Siegel’s theorem on zeros of L-functions | Statement: [Deuring–Heilbronn phenomenon, relatedTo, Siegel’s theorem on zeros of L-functions]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Siegel’s theorem on zeros of L-functions
Context triple: [Deuring–Heilbronn phenomenon, relatedTo, Siegel’s theorem on zeros of L-functions]
  • A. Artin’s conjecture on L-functions
    Artin’s conjecture on L-functions is a major unproven hypothesis in number theory asserting that nontrivial Artin L-functions associated to Galois representations are entire, with deep implications for the distribution of primes and the structure of number fields.
  • B. Euler products for automorphic L-functions
    Euler products for automorphic L-functions are infinite product expansions attached to automorphic representations that encode deep arithmetic information and generalize the classical Euler product of the Riemann zeta function to a broad class of L-functions in the Langlands program.
  • C. Serre’s conjecture on Galois representations
    Serre’s conjecture on Galois representations is a landmark statement in number theory that predicts which two-dimensional mod p Galois representations of the absolute Galois group of the rationals arise from modular forms.
  • D. Ramanujan–Petersson conjecture
    The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
  • E. Birch and Swinnerton-Dyer Conjecture
    The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
  • 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: Siegel’s theorem on zeros of L-functions
Triple: [Deuring–Heilbronn phenomenon, relatedTo, Siegel’s theorem on zeros of L-functions]
Generated description
Siegel’s theorem on zeros of L-functions is a result in analytic number theory that gives strong bounds on how close nontrivial zeros of Dirichlet L-functions can approach 1, with deep implications for the distribution of primes in arithmetic progressions.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Siegel’s theorem on zeros of L-functions
Target entity description: Siegel’s theorem on zeros of L-functions is a result in analytic number theory that gives strong bounds on how close nontrivial zeros of Dirichlet L-functions can approach 1, with deep implications for the distribution of primes in arithmetic progressions.
  • A. Artin’s conjecture on L-functions
    Artin’s conjecture on L-functions is a major unproven hypothesis in number theory asserting that nontrivial Artin L-functions associated to Galois representations are entire, with deep implications for the distribution of primes and the structure of number fields.
  • B. Euler products for automorphic L-functions
    Euler products for automorphic L-functions are infinite product expansions attached to automorphic representations that encode deep arithmetic information and generalize the classical Euler product of the Riemann zeta function to a broad class of L-functions in the Langlands program.
  • C. Serre’s conjecture on Galois representations
    Serre’s conjecture on Galois representations is a landmark statement in number theory that predicts which two-dimensional mod p Galois representations of the absolute Galois group of the rationals arise from modular forms.
  • D. Ramanujan–Petersson conjecture
    The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
  • E. Birch and Swinnerton-Dyer Conjecture
    The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
  • 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_69ca834ca1c88190a11ffb0200342fac completed March 30, 2026, 2:06 p.m.
NER Named-entity recognition batch_69cc479999c881908c0c4e01c07d02d4 completed March 31, 2026, 10:15 p.m.
NED1 Entity disambiguation (via context triple) batch_69cebc42922c819099a464d2e347dec4 completed April 2, 2026, 6:58 p.m.
NEDg Description generation batch_69cec03fa5dc8190bbfe40aa1a3b27c1 completed April 2, 2026, 7:15 p.m.
NED2 Entity disambiguation (via description) batch_69cec0cac51c8190962a23d53c1fb48b completed April 2, 2026, 7:17 p.m.
Created at: March 30, 2026, 6:28 p.m.