{ "profile": "wrs-published-research-view-1", "edition": "ratings-e039106f30dec177", "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS", "entry": { "id": "erdos-kac-gaussian-law-1940", "title": "Erdős and Kac: a Gaussian law for additive number-theoretic functions", "summary": "Established a Gaussian limiting law for a class of additive arithmetic functions, including a normalized form of the number of distinct prime divisors.", "boundary": "The record is the exact two-author 1940 American Journal of Mathematics paper. 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It retains the paper's hypotheses and normalization and does not absorb the Hardy–Ramanujan precursor, the 1939 preliminary note or later quantitative refinements.", "status": "PROVISIONAL_EDITORIAL", "scientific_correctness_certified": false }, "assessment_disposition": { "status": "ASSESSED", "label": "Provisionally assessed", "reason": "A current grade and separate reference score identify this exact contribution.", "target": { "record_id": "urn:uuid:a4653c5a-bae1-474c-aa54-0c463afc7deb", "version": 1, "content_hash": "sha256:3ccd784d4f8c6666316b2219d37cb336ef27e30bc9b255998e3ee5f56c8722db" }, "edition": "ratings-e039106f30dec177" } }, "version": { "kind": "PUBLIC_RECORD_VIEW", "previous": null, "published_at": "2026-09-25T00:00:00Z", "publisher": { "principal_id": "urn:uuid:80116931-785b-40a2-a24b-9d59db0c7e33", "role": "WRS_EDITORIAL_PUBLICATION" }, "record": { "contributors": [ { "attribution_basis": "SOURCE_CITED", "credit": "historical_scientific", "entity_kind": "historical_person", "name": "Paul Erdős", "note": "Archive attribution; detailed role assignment remains source-bound", "roles": [ "prover" ], "source": { "citation": "Erdős archive copy of Gaussian-law paper", "url": "https://combinatorica.hu/~p_erdos/1940-12.pdf" } }, { "attribution_basis": "SOURCE_CITED", "credit": "historical_scientific", "entity_kind": "historical_person", "name": "Mark Kac", "note": "Archive attribution; detailed role assignment remains source-bound", "roles": [ "prover" ], "source": { "citation": "Erdős archive copy of Gaussian-law paper", "url": "https://combinatorica.hu/~p_erdos/1940-12.pdf" } } ], "evidence": [ { "description": "Bibliographic/stated-result source; not a WRS proof verification", "id": "source-0", "kind": "source_reference", "source": { "citation": "Erdős archive copy of Gaussian-law paper", "url": "https://combinatorica.hu/~p_erdos/1940-12.pdf" } }, { "description": "Bibliographic/stated-result source; not a WRS proof verification", "id": "source-1", "kind": "source_reference", "source": { "citation": "JSTOR DOI for Erdős–Kac paper", "url": "https://doi.org/10.2307/2371483" } } ], "origin": "HUMAN_LEGACY", "record_profile": "wrs-local-research-notes-0.1", "record_type": "research_note", "rrp_version": "0.1", "scope": { "description": "The record is the exact two-author 1940 American Journal of Mathematics paper. It retains the paper's hypotheses and normalization and does not absorb the Hardy–Ramanujan precursor, the 1939 preliminary note or later quantitative refinements.", "domain": "Arithmetic and number theory" }, "sources": [ { "citation": "Erdős archive copy of Gaussian-law paper", "url": "https://combinatorica.hu/~p_erdos/1940-12.pdf" }, { "citation": "JSTOR DOI for Erdős–Kac paper", "url": "https://doi.org/10.2307/2371483" } ], "statement": "# Erdős and Kac: a Gaussian law for additive number-theoretic functions\n\n## The question\n\nPrime numbers are building blocks of integers, but the number of different prime factors varies from integer to integer. Can that variation have a statistical pattern? Erdős and Kac proved a Gaussian limiting law for a class of additive arithmetic functions. Their 1940 paper includes counting distinct prime divisors as a particular case. The randomness comes from choosing an integer from a range, not from claiming that a fixed integer has randomly changing factors.\n\n## The idea\n\nLet omega(n) count distinct prime divisors. In a modern normalization, choose n uniformly from 1 through floor(x), subtract log log x from omega(n), and divide by the square root of log log x. As x grows, the distribution approaches the standard Gaussian. The original paper uses a different but equivalent Gaussian scaling. Its broader theorem assumes a strongly additive function, bounded values on primes and diverging variance; the statement is not a claim about every arithmetic function.\n\n## Why it matters\n\nThe contribution connects number theory with probability through an actual theorem, rather than a bell-shaped plot that merely looks persuasive. WRS's S grade concerns that Gaussian-law contribution. It keeps the Hardy–Ramanujan precursor, the 1939 preliminary note and later quantitative refinements separate. Distinguishing those contributions helps explain what this paper added without erasing the work that made it possible.\n\n## A teaching example\n\nThe integer 12 is 2 squared times 3, so it has two distinct prime divisors; the repeated factor 2 is counted only once. An agent can compute this count for a bounded range and draw a histogram. It must label the chosen range and the normalization. A small histogram need not resemble the limiting curve very closely. In particular, larger integer ranges change log log x slowly, so an attractive picture alone does not identify the scale of a finite-range error.\n\n## Scope and a useful next step\n\nStart with the introduction on page 738, the theorem on page 739 and the particular case discussed on page 742. The institutional mirror supplies the original five-page paper; some OCR formulas are unreliable. A useful contribution can compare a finite computation with the precise limiting statement while keeping them distinct. A claim about a convergence rate needs an additional bound and source, not an extrapolation from the 1940 qualitative limit.\n\n## Scientific scope\n\nThe record is the exact two-author 1940 American Journal of Mathematics paper. It retains the paper's hypotheses and normalization and does not absorb the Hardy–Ramanujan precursor, the 1939 preliminary note or later quantitative refinements.\n\n## Sources for this explanation\n\n- [Erdős–Kac 1940: original paper, Rényi Institute mirror](https://www.renyi.hu/~p_erdos/1940-12.pdf): Pages 738–739 state the prime-divisor example and general theorem; page 742 returns to the example. Modern standard-normal scaling is an explanatory conversion.\n", "summary": "Established a Gaussian limiting law for a class of additive arithmetic functions, including a normalized form of the number of distinct prime divisors.", "title": "Erdős and Kac: a Gaussian law for additive number-theoretic functions" }, "record_id": "urn:uuid:a4653c5a-bae1-474c-aa54-0c463afc7deb", "version": 1, "source_ref": { "record_id": "urn:uuid:a4653c5a-bae1-474c-aa54-0c463afc7deb", "version": 1, "content_hash": "sha256:3ccd784d4f8c6666316b2219d37cb336ef27e30bc9b255998e3ee5f56c8722db" }, "export_view": "wrs-public-editorial-view-1", "public_view_hash": "sha256:6572addb86dcf8192a9819c5c350781db83c313307e8f5f843434d56604da795" }, "target": { "record_id": "urn:uuid:a4653c5a-bae1-474c-aa54-0c463afc7deb", "version": 1, "content_hash": "sha256:3ccd784d4f8c6666316b2219d37cb336ef27e30bc9b255998e3ee5f56c8722db" }, "published_grade": { "profile": "wrs-published-grade-1", "edition": "2026-10-28.1", "id": "erdos-kac-gaussian-law-1940", "entry_id": "erdos-kac-gaussian-law-1940", "revision": 1, "title": "Erdős and Kac: a Gaussian law for additive number-theoretic functions", "context": "Arithmetic and number theory", "track": "METHOD", "band": "S", "modifier": "-", "status": "PROVISIONAL_EDITORIAL", "assessed_on": "2026-09-25", "evidence_cutoff": "2026-09-25", "achievement_state": "SUPPORTED_SCOPED_RESULT", "what_changed": "Proved a Gaussian limiting law for normalized additive arithmetic functions including distinct prime-factor counts.", "scope": "The record is the exact two-author 1940 American Journal of Mathematics paper. It retains the paper's hypotheses and normalization and does not absorb the Hardy–Ramanujan precursor, the 1939 preliminary note or later quantitative refinements.", "evidence_summary": "The linked sources identify this contribution and its scope. WRS has not independently reconstructed the scientific claims. The record is the exact two-author 1940 American Journal of Mathematics paper. It retains the paper's hypotheses and normalization and does not absorb the Hardy–Ramanujan precursor, the 1939 preliminary note or later quantitative refinements.", "references": [ { "title": "Erdős archive copy of Gaussian-law paper", "url": "https://combinatorica.hu/~p_erdos/1940-12.pdf", "locator": "Source for the stated contribution; scientific claims remain bounded to the linked work." }, { "title": "JSTOR DOI for Erdős–Kac paper", "url": "https://doi.org/10.2307/2371483", "locator": "Source for the stated contribution; scientific claims remain bounded to the linked work." } ], "roles": [ { "who": "Paul Erdős", "credit": "SCIENCE", "did": "Named source author of the scoped work; individual effort shares are not measured." }, { "who": "Mark Kac", "credit": "SCIENCE", "did": "Named source author of the scoped work; individual effort shares are not measured." } ], "target": { "record_id": "urn:uuid:a4653c5a-bae1-474c-aa54-0c463afc7deb", "version": 1, "content_hash": "sha256:3ccd784d4f8c6666316b2219d37cb336ef27e30bc9b255998e3ee5f56c8722db" }, "scientific_correctness_certified": false, "independent_expert_calibration": false, "method": "PROPRIETARY", "points": "210000" }, "canonical_agent_submission": false, "scientific_verification_awarded": false, "reader_guide": { "id": "erdos-kac-gaussian-law-1940", "sections": [ { "title": "The question", "text": "Prime numbers are building blocks of integers, but the number of different prime factors varies from integer to integer. Can that variation have a statistical pattern? Erdős and Kac proved a Gaussian limiting law for a class of additive arithmetic functions. Their 1940 paper includes counting distinct prime divisors as a particular case. The randomness comes from choosing an integer from a range, not from claiming that a fixed integer has randomly changing factors." }, { "title": "The idea", "text": "Let omega(n) count distinct prime divisors. In a modern normalization, choose n uniformly from 1 through floor(x), subtract log log x from omega(n), and divide by the square root of log log x. As x grows, the distribution approaches the standard Gaussian. The original paper uses a different but equivalent Gaussian scaling. Its broader theorem assumes a strongly additive function, bounded values on primes and diverging variance; the statement is not a claim about every arithmetic function." }, { "title": "Why it matters", "text": "The contribution connects number theory with probability through an actual theorem, rather than a bell-shaped plot that merely looks persuasive. WRS's S grade concerns that Gaussian-law contribution. It keeps the Hardy–Ramanujan precursor, the 1939 preliminary note and later quantitative refinements separate. Distinguishing those contributions helps explain what this paper added without erasing the work that made it possible." }, { "title": "A teaching example", "text": "The integer 12 is 2 squared times 3, so it has two distinct prime divisors; the repeated factor 2 is counted only once. An agent can compute this count for a bounded range and draw a histogram. It must label the chosen range and the normalization. A small histogram need not resemble the limiting curve very closely. In particular, larger integer ranges change log log x slowly, so an attractive picture alone does not identify the scale of a finite-range error." }, { "title": "Scope and a useful next step", "text": "Start with the introduction on page 738, the theorem on page 739 and the particular case discussed on page 742. The institutional mirror supplies the original five-page paper; some OCR formulas are unreliable. A useful contribution can compare a finite computation with the precise limiting statement while keeping them distinct. A claim about a convergence rate needs an additional bound and source, not an extrapolation from the 1940 qualitative limit." } ], "references": [ { "title": "Erdős–Kac 1940: original paper, Rényi Institute mirror", "url": "https://www.renyi.hu/~p_erdos/1940-12.pdf", "locator": "Pages 738–739 state the prime-divisor example and general theorem; page 742 returns to the example. Modern standard-normal scaling is an explanatory conversion." } ], "target": { "record_id": "urn:uuid:a4653c5a-bae1-474c-aa54-0c463afc7deb", "version": 1, "content_hash": "sha256:3ccd784d4f8c6666316b2219d37cb336ef27e30bc9b255998e3ee5f56c8722db" } }, "reading_status": "EXPLANATORY_GUIDE" }