{"id":"34a7cf6f-c82e-41c0-b6d3-3ab86d20e506","arxiv_id":"1907.07664","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The inequality log g(n) < li^{-1}(n) for all n > 0 is equivalent to the Riemann hypothesis.","lead":"This paper claims that an inequality involving the Landau function g(n) and the inverse logarithmic integral is equivalent to the Riemann hypothesis. A smart generalist might read it because it offers a potential new characterization of one of the most important open problems in mathematics.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"Growth rates mismatch: log g(n) ∼ √(n log n) while li^{-1}(n) ∼ n log n, so inequality holds unconditionally","rationale":"The reader's weakest assumption directly identifies the risk that the inequality is not precisely equivalent to RH. The growth-rate comparison exposes exactly that gap: one side of the biconditional holds unconditionally, rendering the central claim internally inconsistent with known facts about g(n).","tokens_in":1565,"tokens_out":398,"duration_ms":38544,"concrete_test":"For N=10^6, numerically invert li(x) to obtain li^{-1}(N) and compare to a concrete upper bound on log g(N) obtained by maximizing ∏ p_i^{a_i} subject to ∑ a_i p_i ≤ N (using primes up to ~√(2N log N)); if li^{-1}(N) exceeds the bound by a factor >5, the inequality is non-binding for large n and equivalence fails.","verdict_should_be":"REJECT","load_bearing_attack":"The claimed equivalence requires that ∀n>0, log g(n) < li^{-1}(n) holds if and only if RH is true. Unconditionally, log g(n) ∼ √(n log n) (Landau 1909; see also standard bounds via maximal prime-power partitions). Meanwhile li^{-1}(n), the inverse of li(x) = ∫_2^x dt/log t, satisfies li^{-1}(n) ∼ n log n. Thus log g(n) = o(li^{-1}(n)) for large n, so the inequality is true for all sufficiently large n independently of RH. This violates the 'exactly coextensive' requirement in the reader's weakest assumption; the combination of asymptotics for g(n) and li^{-1} cannot produce an RH-equivalent statement without an order-of-magnitude error in the claimed bound.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript claims that the property 'For all n > 0, log g(n) < li^{-1}(n)' (with g(n) the Landau function) is equivalent to the Riemann hypothesis.","tokens_in":1750,"tokens_out":196,"duration_ms":12778,"significance":"If the claimed equivalence were valid, it would provide a novel characterization of the Riemann hypothesis in terms of the maximal order of elements in the symmetric group S_n.","major_comments":[{"comment":"Abstract: the claimed equivalence is incompatible with the known asymptotics log g(n) ∼ √(n log n) (Landau 1909) and li^{-1}(n) ∼ n log n; the inequality therefore holds unconditionally for all sufficiently large n, so the two sides cannot be coextensive.","section":"Abstract"}],"minor_comments":[],"recommendation":"reject","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for identifying a fundamental inconsistency between the claimed equivalence and the known asymptotics of the functions involved. We address the point directly below.","responses":[{"response":"We agree with the referee. The asymptotic relations log g(n) ∼ √(n log n) and li^{-1}(n) ∼ n log n imply that log g(n) = o(li^{-1}(n)) as n → ∞, so the inequality log g(n) < li^{-1}(n) holds for all sufficiently large n independently of the Riemann hypothesis. The statement that the inequality holds for every n > 0 is therefore equivalent only to a finite verification up to some bound and cannot be coextensive with the Riemann hypothesis. This renders the main claim of the manuscript incorrect as stated.","revision_made":"yes","referee_comment":"[Abstract] Abstract: the claimed equivalence is incompatible with the known asymptotics log g(n) ∼ √(n log n) (Landau 1909) and li^{-1}(n) ∼ n log n; the inequality therefore holds unconditionally for all sufficiently large n, so the two sides cannot be coextensive."}],"tokens_in":1040,"tokens_out":264,"duration_ms":23399,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The central claim here is that the inequality log g(n) < li^{-1}(n) for all n > 0 is equivalent to the Riemann hypothesis. That cannot be correct. The Landau function satisfies log g(n) ~ sqrt(n log n) from the maximal prime-power partitions, while li^{-1}(n) ~ n log n, so the inequality holds unconditionally for all sufficiently large n. This makes the two sides of the claimed equivalence independent, which breaks the result on basic grounds. The abstract states the equivalence without any derivation, lemmas, or calculations to support it, and the growth-rate mismatch is visible from the definitions alone. On the positive side, the paper accurately recalls the two standard characterizations of g(n): the maximal order of an element in S_n and the largest product of prime powers summing to at most n. Those facts are standard and correctly stated. The soft spot is the load-bearing equivalence itself; everything else is background. No information is given on whether the argument uses self-referential estimates or external bounds, but the asymptotics already show the claimed coextensiveness fails. This paper would interest readers hunting for new combinatorial statements equivalent to RH. A reader who wants to check whether any such link via the symmetric group survives scrutiny might look at it, but the mismatch means it does not repay the time. I would not bring it to a reading group. I would not cite it. It does not deserve peer review.","headline":"The claimed equivalence cannot hold: the inequality is true for large n by standard asymptotics on g(n) and li^{-1}(n).","tokens_in":2183,"tokens_out":366,"would_cite":false,"duration_ms":18610,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Landau function / RH equivalence via li^{-1} asymptotics is orthogonal to RS forcing chain","alignment":"orthogonal","rationale":"The paper's core machinery (Landau g(n) as max order in S_n via prime-power partitions, explicit superchampion enumeration, effective Chebyshev bounds, and equivalence of log g(n) < sqrt(li^{-1}(n)) to RH via Omega results on zeta zeros) operates entirely in analytic number theory. It uses the known asymptotic log g(n) ~ sqrt(n log n) matching sqrt(li^{-1}(n)) and conditional error terms, with heavy computation on C1-superchampions. No J-cost, reciprocal symmetry, phi-ladder spacings, 8-tick periodicity, or parameter-free constant derivation appears. RS theorems (reality_from_one_distinction, J-uniqueness via Aczel, AlexanderDuality D=3, etc.) neither confirm nor contradict the claimed RH equivalence.","tokens_in":87650,"confidence":"high","tokens_out":209,"duration_ms":7796,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The inequality log g(n) < li^{-1}(n) for all positive integers n holds exactly when the Riemann hypothesis is true.","keywords":["Landau function","Riemann hypothesis","logarithmic integral","symmetric group","maximal order","prime powers","equivalence"],"falsifier":"An explicit integer n at which the computed value of log g(n) meets or exceeds li^{-1}(n) would violate the inequality and, by the claimed equivalence, falsify the Riemann hypothesis.","tokens_in":2468,"feed_emoji":"","tokens_out":698,"duration_ms":19156,"temperature":0.7,"pith_summary":"This paper proves an equivalence between the Riemann hypothesis and the statement that log g(n) stays strictly below the inverse logarithmic integral li^{-1}(n) for every positive integer n. The Landau function g(n) is the largest order of any permutation of n objects, realized as the biggest product of prime powers whose weighted sum is at most n. The comparison uses the inverse of the logarithmic integral, a function that encodes the prime-number counting behavior central to the zeta function. A reader cares because the result converts one of the most famous unsolved problems into a concrete, in-principle checkable inequality involving a single arithmetic function.","feed_headline":"Landau function inequality equivalent to Riemann hypothesis","feed_subtitle":"The bound log g(n) < li^{-1}(n) for all n holds exactly when the Riemann hypothesis is true.","key_machinery":"The Landau function g(n), the largest product of prime powers with sum of the terms at most n, together with its direct comparison against the inverse logarithmic integral li^{-1}(n).","core_discovery":"The main result is that the property 'For all n > 0, log g(n) < li^{-1}(n)' (where g(n) is the maximal order of an element of the symmetric group of degree n) is equivalent to the Riemann hypothesis.","pith_inferences":["Direct computation of g(n) for successively larger n supplies a practical route to test the inequality numerically up to bounds where li^{-1}(n) can still be evaluated accurately.","The equivalence supplies a new arithmetic-function formulation that could be compared with other known reformulations of the Riemann hypothesis for possible simplifications or contradictions.","If the inequality holds up to a computable limit, it would confirm the hypothesis inside the corresponding range of zeta zeros, though this remains a finite check."],"forward_implications":["The Riemann hypothesis holds if and only if the inequality log g(n) < li^{-1}(n) is true for every positive integer n.","Any counterexample n to the inequality would immediately disprove the Riemann hypothesis.","Asymptotic and extremal results already known for g(n) translate directly into statements about the location of zeta zeros.","The growth of the maximal orders in symmetric groups is governed by the same prime-distribution law that appears in the Riemann hypothesis."],"fun_headline_variants":["Landau g(n) bound equivalent to Riemann hypothesis","Riemann hypothesis equivalent to Landau g(n) bound","log g(n) bound equivalent to Riemann hypothesis","Landau function bound equivalent to Riemann hypothesis"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The known asymptotic and extremal properties of the Landau function g(n), when combined with the definition of li^{-1}, produce an inequality whose validity is exactly coextensive with the Riemann hypothesis.","fun_headline_variants_meta":{"raw":{"variants":["Landau g(n) bound equivalent to Riemann hypothesis","Riemann hypothesis equivalent to Landau g(n) bound","log g(n) bound equivalent to Riemann hypothesis","Landau function bound equivalent to Riemann hypothesis"]},"model":"grok-4.3","cost_usd":0.006946,"raw_usage":{"total_tokens":3133,"prompt_tokens":494,"num_sources_used":0,"completion_tokens":58,"cost_in_usd_ticks":69462000,"prompt_tokens_details":{"text_tokens":494,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2581,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":494,"tokens_out":58,"duration_ms":16198,"temperature":1.0,"reasoning_tokens":2581,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-24T20:05:47.749882+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit integer n at which the computed value of log g(n) meets or exceeds li^{-1}(n) would violate the inequality and, by the claimed equivalence, falsify the Riemann hypothesis.","supporting_citations":[],"review_version":1}