{ "model": { "rank": 18552, "code": "thing-q620595", "model_id": "vr.tr.dirichlet-series", "name": "Dirichlet series", "purpose": "Let an agent explain Dirichlet series by definition, convergence, examples and role in number theory.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.REL", "industry": "", "version": "", "url": "/models/thing/q620595/", "tier": 2, "score": 29, "payload": { "layer": "wikidata", "aliases": [], "aliasCount": 0, "merged": 0, "knownIn": 29, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q620595", "url": "https://www.wikidata.org/wiki/Q620595", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-11T14:09:48Z", "providers": [ "Claude" ], "breadth": "written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify", "pass": 2, "wave": 1, "engine": "claude" }, "metaModel": { "id": "THING-Q620595", "registryId": "vr.tr.dirichlet-series", "name": "Dirichlet series", "version": "0.2.0-wave.1", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.REL" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q620595/", "model": { "registry_id": "vr.tr.dirichlet-series", "name": "Dirichlet series", "purpose": "Let an agent explain Dirichlet series by definition, convergence, examples and role in number theory.", "definition": "An infinite series of the form sum of a(n) divided by n to the power s, where s is a complex variable, of which the Riemann zeta function is the simplest example; Dirichlet series converge in half-planes, have Euler products when coefficients are multiplicative, and underlie L-functions in analytic number theory.", "what_it_is_for": "Analytic number theory and related mathematics.", "affordances": [ "identify a Dirichlet series", "find its half-plane of convergence", "derive Euler products for multiplicative coefficients", "connect series to arithmetic functions" ], "distinguishing_features": [ "Coefficients over n to the minus s", "Half-plane convergence", "Euler products", "Encode arithmetic functions" ], "appearance": "Not physical; sums over n with terms a(n) n^(-s).", "visual_identification": [ "Sums of a(n)/n^s", "Riemann zeta and L-functions", "Power series use powers of a variable instead" ], "physical_properties": [], "families_and_kinds": [ "Riemann zeta function", "Dirichlet L-functions", "series of arithmetic functions", "general Dirichlet series" ], "related_models": [ { "relation": "is a kind of", "target": "general Dirichlet series", "why": "category" }, { "relation": "includes", "target": "Riemann zeta function", "why": "example" }, { "relation": "is related to", "target": "alternating series", "why": "series theory" }, { "relation": "is studied in", "target": "analytic number theory", "why": "field" } ], "identifiers": [], "standards_and_regulation": [ "No specific regulation" ], "failure_modes_and_hazards": [ "Misstating convergence regions", "Assuming Euler products for non-multiplicative coefficients" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "definition", "name": "Definition", "description": "What it is.", "rationale": "Form defines the series.", "layers": [ { "id": "form", "name": "Form", "description": "General form.", "findings": [ { "id": "form-finding", "name": "Form", "description": "General form.", "questions": [ { "text": "Is this series of the form sum a(n) n^(-s)?", "kind": "boundary" }, { "text": "What are its coefficients?", "kind": "definition" } ] } ] }, { "id": "examples", "name": "Examples", "description": "Classic examples.", "findings": [ { "id": "examples-finding", "name": "Examples", "description": "Examples.", "questions": [ { "text": "How do the zeta function and Dirichlet L-functions arise as Dirichlet series?", "kind": "definition" }, { "text": "Which arithmetic functions give other examples?", "kind": "definition" } ] } ] } ] }, { "id": "convergence", "name": "Convergence", "description": "Where it converges.", "rationale": "Half-planes matter.", "layers": [ { "id": "abscissa", "name": "Abscissa", "description": "Abscissa of convergence.", "findings": [ { "id": "abscissa-finding", "name": "Abscissa", "description": "Abscissa of convergence.", "questions": [ { "text": "What are the abscissas of convergence and absolute convergence?", "kind": "measurement" }, { "text": "How are they found?", "kind": "definition" } ] } ] }, { "id": "continuation", "name": "Continuation", "description": "Analytic continuation.", "findings": [ { "id": "continuation-finding", "name": "Continuation", "description": "Analytic continuation.", "questions": [ { "text": "Can the function be continued beyond its half-plane of convergence?", "kind": "definition" }, { "text": "What is known about its poles?", "kind": "provenance" } ] } ] } ] }, { "id": "structure", "name": "Structure", "description": "Euler products.", "rationale": "Multiplicativity helps.", "layers": [ { "id": "euler", "name": "Euler", "description": "Euler products.", "findings": [ { "id": "euler-finding", "name": "Euler", "description": "Euler products.", "questions": [ { "text": "When does a Dirichlet series have an Euler product?", "kind": "definition" }, { "text": "What is the product here?", "kind": "measurement" } ] } ] }, { "id": "multiplication", "name": "Multiplication", "description": "Dirichlet convolution.", "findings": [ { "id": "multiplication-finding", "name": "Multiplication", "description": "Dirichlet convolution.", "questions": [ { "text": "How does multiplying Dirichlet series correspond to Dirichlet convolution of coefficients?", "kind": "definition" }, { "text": "Which identities follow?", "kind": "definition" } ] } ] } ] }, { "id": "learning", "name": "Learning", "description": "Study.", "rationale": "References help.", "layers": [ { "id": "references", "name": "References", "description": "Textbooks.", "findings": [ { "id": "references-finding", "name": "References", "description": "References.", "questions": [ { "text": "Which textbooks introduce Dirichlet series and L-functions?", "kind": "provenance" }, { "text": "What prerequisites are needed?", "kind": "definition" } ] } ] }, { "id": "open", "name": "Open problems", "description": "Research questions.", "findings": [ { "id": "open-finding", "name": "Open problems", "description": "Open problems.", "questions": [ { "text": "Which open problems, such as the Riemann hypothesis, involve Dirichlet series?", "kind": "provenance" }, { "text": "Is the status reported accurately?", "kind": "boundary" } ] } ] } ] } ] }, "openQuestions": [ "Should L-functions be separate entries?", "How should standard references be linked?", "How should examples be linked?" ], "statistics": { "bundles": 4, "layers": 8, "findings": 8, "questions": 16 } }, "draft": { "generator": "vr.draft.v3", "status": "draft-generated", "researched": false, "archetype": "abstract concept", "method": "Written from the archetype playbook - what this kind of thing needs beyond identity and provenance - and from the structure that recurred across 6,333 models already researched by two engines. Applied to this entry by rule. No source was read for this thing and no claim here is researched. This entry carries no facets of its own, so they were inferred from its domain - a guess about a whole domain applied to one thing.", "facetsInferred": true, "nextPass": "A researcher replaces this draft with a sourced specification. Treat every sentence below as a proposal to argue with.", "purpose": "Give an agent a durable, checkable way to recognise a Dirichlet series, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain Dirichlet series by definition, convergence, examples and role in number theory.", "characteristics": { "substance": "abstract", "origin": "conceptual", "agency": "inert" }, "whatYouCanDoWithIt": [ "observed and measured" ], "distinguishingFeatures": [ "Described in 29 Wikipedia languages, which is a measure of how widely the thing is known, not of how important it is." ], "openQuestionsForResearch": [ "Which of the bundles below does a real task actually need, and which are ceremony?", "What does this thing have that the facets do not capture at all?", "Which neighbouring kind is most often confused with a Dirichlet series, and on what evidence are they told apart?" ], "whatItIsMadeOf": "an abstraction with no physical instance", "physicalCharacter": [ "Does nothing on its own; everything it does, something else did to it.", "These come from the domain this entry sits in rather than from the entry itself, so treat them as a first guess about the whole domain applied to one thing." ], "whatCanBeDoneWithIt": [ "observe it, measure it, record its state" ], "howItIsRecognised": [ "Nothing to see. What is recognised is an instance of it, and which instances count is exactly what is argued about." ], "relatedModels": [], "standing": "Described in 29 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is.", "structure": { "bundles": [ { "id": "identity-and-classification", "name": "Identity, naming and classification", "description": "How an agent tells one Dirichlet series from another, and a Dirichlet series from things that resemble it.", "rationale": "Recognition comes before every other claim. Without stable identity nothing else in the model can be trusted to be about the same thing twice.", "layers": [ { "id": "naming-and-identifiers", "name": "Names and identifiers", "description": "The names this thing goes by and the identifiers that survive translation and time.", "findings": [ { "id": "preferred-name-and-aliases", "name": "Preferred name, aliases and local names", "description": "Which name to use, which names mean the same thing, and which merely sound similar.", "questions": [ { "id": "preferred-name-and-aliases-q01", "text": "What identifies and describes the name of a Dirichlet series, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "preferred-name-and-aliases-q02", "text": "Who or what asserted this about the name of a Dirichlet series, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "preferred-name-and-aliases-q03", "text": "What may an agent decide or do once the name of a Dirichlet series is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "stable-identifiers", "name": "Stable identifiers and external keys", "description": "Identifiers that keep pointing at this kind of thing across systems and languages.", "questions": [ { "id": "stable-identifiers-q01", "text": "What identifies and describes an identifier for a Dirichlet series, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stable-identifiers-q02", "text": "Who or what asserted this about an identifier for a Dirichlet series, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stable-identifiers-q03", "text": "What may an agent decide or do once an identifier for a Dirichlet series is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "classification-and-granularity", "name": "Classification and granularity", "description": "Where a Dirichlet series sits among kinds, and how finely a task needs to cut it.", "findings": [ { "id": "kind-and-parents", "name": "Kind, parents and neighbouring kinds", "description": "The classes this thing belongs to and the ones it is next to.", "questions": [ { "id": "kind-and-parents-q01", "text": "What identifies and describes the kind of a Dirichlet series, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "kind-and-parents-q02", "text": "Who or what asserted this about the kind of a Dirichlet series, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "kind-and-parents-q03", "text": "What may an agent decide or do once the kind of a Dirichlet series is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "distinguishing-features", "name": "Distinguishing features", "description": "What separates a Dirichlet series from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a Dirichlet series, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "distinguishing-features-q02", "text": "Who or what asserted this about what distinguishes a Dirichlet series, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "distinguishing-features-q03", "text": "What may an agent decide or do once what distinguishes a Dirichlet series is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "state-and-lifecycle", "name": "State and lifecycle", "description": "The states a Dirichlet series passes through and the events that move it between them.", "rationale": "Most decisions about a thing depend on what state it is in now, which is a claim with a time on it, not a property.", "layers": [ { "id": "lifecycle-stages", "name": "Lifecycle stages", "description": "From coming into existence to ceasing to be one of these.", "findings": [ { "id": "stages-and-transitions", "name": "Stages and transitions", "description": "The stages worth naming and what moves a Dirichlet series between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a Dirichlet series, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stages-and-transitions-q02", "text": "Who or what asserted this about the lifecycle of a Dirichlet series, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stages-and-transitions-q03", "text": "What may an agent decide or do once the lifecycle of a Dirichlet series is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "observations-and-status", "name": "Observations and current status", "description": "What is observed about a Dirichlet series, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a Dirichlet series is recorded so that it can be superseded rather than overwritten.", "questions": [ { "id": "observation-record-q01", "text": "What identifies and describes an observation of a Dirichlet series, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "observation-record-q02", "text": "Who or what asserted this about an observation of a Dirichlet series, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "observation-record-q03", "text": "What may an agent decide or do once an observation of a Dirichlet series is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "definitions-in-contest", "name": "Definitions and who holds them", "description": "What Dirichlet series is taken to mean, and by whom.", "rationale": "When a field disagrees about a concept, the disagreement is the content. A model that picks one definition silently destroys the information.", "layers": [ { "id": "competing-definitions", "name": "Competing definitions", "description": "The main readings and the traditions behind them.", "findings": [ { "id": "definition-map", "name": "Definitions and their holders", "description": "Each definition with the school or body that holds it.", "questions": [ { "id": "definition-map-q01", "text": "Which definitions of Dirichlet series are in use, and which tradition or body holds each?", "kind": "definition" }, { "id": "definition-map-q02", "text": "What turns on the difference between them in practice?", "kind": "boundary" } ] } ] }, { "id": "operationalisation", "name": "Operationalisation", "description": "How it is measured or applied when it has to be.", "findings": [ { "id": "operational-record", "name": "Measures and proxies", "description": "Instruments and indicators used to stand in for it.", "questions": [ { "id": "operational-record-q01", "text": "How is Dirichlet series operationalised or measured in practice, and by what instrument?", "kind": "measurement" }, { "id": "operational-record-q02", "text": "What does that operationalisation leave out, and when does that matter?", "kind": "boundary" } ] } ] } ] }, { "id": "instances-and-use", "name": "Instances, use and consequence", "description": "What counts as an instance of Dirichlet series and what follows from calling something that.", "rationale": "Applying a concept is an act with consequences, so a model must say what the label licenses and what it does not.", "layers": [ { "id": "instances", "name": "What counts as an instance", "description": "Clear cases, borderline cases and non-cases.", "findings": [ { "id": "instance-tests", "name": "Tests for an instance", "description": "What would settle whether something falls under it.", "questions": [ { "id": "instance-tests-q01", "text": "What would settle whether something is an instance of Dirichlet series?", "kind": "boundary" }, { "id": "instance-tests-q02", "text": "Which borderline cases are argued about, and on what grounds?", "kind": "definition" } ] } ] }, { "id": "consequence", "name": "Consequence of application", "description": "Rights, duties or decisions that follow from the label.", "findings": [ { "id": "consequence-record", "name": "What the label licenses", "description": "What an agent may do once something is classified this way.", "questions": [ { "id": "consequence-record-q01", "text": "What follows practically once something is treated as Dirichlet series?", "kind": "action" }, { "id": "consequence-record-q02", "text": "What must an agent not infer from the label alone?", "kind": "action" } ] } ] } ] }, { "id": "provenance-and-evidence", "name": "Provenance, evidence and time", "description": "Where every claim about a Dirichlet series came from and when it held.", "rationale": "A claim without a source and a time cannot be superseded, only overwritten, and an agent that overwrites loses the ability to explain itself.", "layers": [ { "id": "source-and-authority", "name": "Source and authority", "description": "Who said it, on what evidence, and how strongly.", "findings": [ { "id": "claim-provenance", "name": "Claim provenance and confidence", "description": "The authority behind each claim about a Dirichlet series and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a Dirichlet series, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "claim-provenance-q02", "text": "Who or what asserted this about a claim about a Dirichlet series, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "claim-provenance-q03", "text": "What may an agent decide or do once a claim about a Dirichlet series is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "time-and-versions", "name": "Time, versions and supersession", "description": "When a claim was true, when it was learnt, and what replaced it.", "findings": [ { "id": "validity-and-supersession", "name": "Validity period and supersession", "description": "How an old claim about a Dirichlet series is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a Dirichlet series, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "validity-and-supersession-q02", "text": "Who or what asserted this about the validity of a claim about a Dirichlet series, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "validity-and-supersession-q03", "text": "What may an agent decide or do once the validity of a claim about a Dirichlet series is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] } ] }, "statistics": { "bundles": 5, "layers": 10, "findings": 12, "questions": 32 } } } }