{"id":"b00ee224-fcd3-44f2-8d44-f76682a330d5","arxiv_id":"1908.00488","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Tenenbaum surveys progress on Erdős' problems about divisors, sets of multiples, and prime factors, and proves that the distribution of τ+(n)/τ(n) is continuous at 1.","lead":"This survey revisits a 1979 Erdős problem list on divisors and multiples and records what is now known about each question. It is a useful map of results in analytic number theory, and it includes a small new theorem on the distribution of close divisors.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the proof of Theorem 1 is a sound sketch given the cited standard results.","rationale":"The reader's weakest_assumption identified the reliance on external lemmas (Theorem 51 and Lemma 48.1 of [46]). My stress-test examined whether this reliance is actually load-bearing in a problematic way. The proof of Theorem 1 is a sketch, but it is coherent: the two cited results supply, respectively, a close pair of small divisors for almost all n and a discrepancy bound for the rough-part divisor set. I verified the counting step: for each good m, the pair (md, md′) occupies one dyadic interval, and grouping by interval still yields a saving of one extra divisor per good m, so the key inequality holds. The subsequent bound on τ(n_ε) is standard. The continuity conclusion follows by epsilon–delta. I found no internal gap. The only potential concern is that Lemma 48.1's exact hypotheses are not restated, but the text explicitly applies it to m | (n/n_ε), so there is no apparent misapplication. This is a verification issue rather than a substantive flaw, so the verdict remains unchanged.","tokens_in":21294,"tokens_out":24504,"duration_ms":250201,"concrete_test":"Check the statement of Lemma 48.1 in Hall–Tenenbaum, Divisors (1988), and verify that it bounds the discrepancy of {(log m)/log 2 : m | N} for N = n/n_ε (the T_ε-rough part) on a set of n of lower density at least 1−ε/3, with the discrepancy bound ε. If the lemma is only stated for all divisors of n, re-derive the rough-part version from the book's methods; if it cannot be extended, Theorem 1 would need a different proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After close reading, I find no internal inconsistency or misapplication in the proof of Theorem 1. The argument correctly combines Theorem 51 of Hall–Tenenbaum (a quantitative close-divisors result, presumably with ratio 2^ε, as the notation '2εd' is an OCR artifact) and Lemma 48.1, which the text explicitly states bounds the discrepancy of {(log m)/log 2 : m | (n/n_ε)} on a set of n of lower density 1−ε/3. For each good m, the pair (md, md′) lies in one dyadic interval; even if several m map to the same interval, the union of their pairs contributes at least one extra divisor per m, so τ_+(n) ≤ τ(n) − (1−ε)τ(n/n_ε) is valid. The bound τ(n_ε) ≤ log T_ε for almost all n is standard since n_ε is smooth with typically small divisor count. The epsilon–delta bookkeeping then correctly yields lim_{z→1−} ν(z) = 1. The only residual risk is a verification issue: Lemma 48.1 is cited but not restated, and one could worry it applies to the full divisor set rather than to the T_ε-rough part. The text explicitly says it applies to m | (n/n_ε), and the author is a co-author of the cited book, so this is not a substantive flaw.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper is a historical survey of Erdős's 1979 article [24] on unconventional problems in number theory, thirty-four years after its publication. The author updates the status of each problem, describes descendants and posterity, and includes two original results: Theorem 1 proves that the limiting distribution ν(z) of τ_+(n)/τ(n) is continuous at z = 1, and Theorem 2 gives a direct proof that an integer sequence with logarithmic density 1 has a set of multiples with natural density 1. The survey covers close divisors, the Erdős–Hooley Delta function, Behrend sequences, divisor statistics, and problems on largest prime factors of consecutive integers, with extensive references.","tokens_in":21557,"tokens_out":22654,"duration_ms":196624,"significance":"If correct, Theorem 1 answers the endpoint part of an open problem raised in the paper, namely the determination of discontinuity points of ν. Theorem 2 provides an elementary self-contained proof of a statement that follows from the Davenport–Erdős theorem, and the proof is interesting in its own right. As a survey, the paper is authoritative and unusually informative: it is written by a leading contributor to the subject and includes many recent results with precise statements. The author is careful to point out limitations, notably in footnote 3, where he notes that Theorem 1 does not yield a new proof of Erdős's conjecture (9) because a refinement of (9) was used. The survey statements appear accurate, and the proofs of the two theorems are plausible; Theorem 2's proof is self-contained.","major_comments":[],"minor_comments":[{"comment":"The proof relies on Theorem 51 and Lemma 48.1 of [46] without stating them; please include their precise statements or at least a clear description so that the proof can be checked without the book. Also fix the notation '2εd' to '2^ε d' and 'T ε' to 'T_ε'.","section":"Proof of Theorem 1"},{"comment":"In the same proof, specify that the discrepancy in Lemma 48.1 refers to the fractional parts {(log m)/log 2} modulo 1, since the argument depends on distances to integers.","section":"Proof of Theorem 1"},{"comment":"The assertion 'τ(n_ε) ≤ log T_ε holds on a sequence of lower density 1 − ε/3' is stated without proof or reference; please add a brief justification or a citation to a standard result on smooth parts of integers.","section":"Proof of Theorem 1"},{"comment":"Footnote 3 correctly notes that Theorem 1 does not provide a new proof of (9) because a refinement of (9) was used; this circularity is acknowledged, but it would help to also mention this dependency near the statement of Theorem 1 so readers are not misled.","section":"Footnote 3"},{"comment":"There are numerous OCR/typographical errors (e.g., '3 ω (n)' for '3^{ω(n)}', 'greaterorequalslant' for '≥', 'd, d, d' for the three density symbols); a careful proofreading pass would improve readability.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"The paper is a solid survey with a credible new result in Theorem 1. The proof of Theorem 1 is concise and relies on two lemmas from the author's book; I would like to see those lemmas stated for completeness. No major mathematical obstacle was found. The paper fits the journal's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague—\n\nThe paper is a survey with one small new result, and the new result is the part that needs attention. Theorem 1 proves that the limiting distribution ν of τ+(n)/τ(n) is continuous at z = 1, which answers an open question Tenenbaum had flagged. It is a modest step, not a breakthrough, but it is real. The proof is a sketch: it invokes Theorem 51 and Lemma 48.1 from Hall–Tenenbaum, Divisors, without restating them. I traced the argument; it works. The stress-test note is right that Lemma 48.1 is applied to m | (n/n_ε), exactly as the text says, and since Tenenbaum co-wrote the source, the risk of misapplication is low. Still, a referee would reasonably ask for the lemmas to be stated, because the paper alone doesn't give a reader enough to verify the step.\n\nThe survey portion is genuinely useful. It updates Erdős's 1979 problems on close divisors, Hooley's Delta-function, Behrend sequences, the k-th prime factor, and largest prime factors of consecutive integers. Tenenbaum distinguishes proved results from conjectures and flags where Erdős's original guesses were off. The historical thread—Erdős's five criteria for a good conjecture—gives structure without becoming filler. Theorem 2 is not new; it is a direct proof of a known corollary of Davenport–Erdős, but it is clean and a nice self-contained addition.\n\nThe heavy self-citation is appropriate here. Tenenbaum is the main contributor in several of these areas, and the cited results are published, not being used to prove themselves. The footnote admitting that Theorem 1 does not yield a new proof of the close-divisors conjecture is honest and should stay.\n\nSoft spots: the proof of Theorem 1 is too compressed for a reader to verify without going to the book, and the survey is more of a guided tour than a critical map—you'll have to assemble the list of open problems yourself. Minor. The mathematics itself checks out.\n\nWho gets value: number theorists working on divisors, multiplicative structure, or Behrend sequences. It's a reference-grade survey with a small new result. Send it to a serious referee, with a request to expand Theorem 1's proof; after that I'd be comfortable seeing it published.","headline":"A genuine but small new result inside a useful survey; the proof of Theorem 1 is too sketchy to verify from the page alone, but the paper deserves referee time.","tokens_in":22074,"tokens_out":3440,"would_cite":true,"duration_ms":34797,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":null,"created_at":"2026-08-14T15:52:15.006581+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":null,"supporting_citations":[],"review_version":1}