{"id":"72676c7b-d2ee-4587-b964-42aebd24da1c","arxiv_id":"2605.29111","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":8.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Constructs a counterexample 1-bounded completely multiplicative function disproving Goldmakher's 2009 conjecture on the boundedness of logarithmically-averaged partial sums.","lead":"This paper constructs a 1-bounded completely multiplicative function f making the limsup of its log-averaged partial sums over 1 plus exp of the real-part sum over primes go to infinity. A smart generalist might read it because it shows how explicit constructions can disprove longstanding conjectures in analytic number theory.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption matches the construction step that the full text now supplies. With that step verified as explicit and the estimates standard, the disproof stands as stated; the prior UNVERDICTED status was solely due to abstract-only access.","tokens_in":1588,"tokens_out":238,"duration_ms":16251,"concrete_test":"Extract the explicit rule for f(p) from the construction section; recompute the ratio numerically along the sequence of x given in the proof up to x = 10^12 (or the largest feasible); confirm the ratio exceeds 100.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The manuscript supplies an explicit construction of the completely multiplicative f with |f(n)| ≤ 1 for all n (via values on primes satisfying |f(p)| ≤ 1). The argument then directly exhibits a sequence of x where the numerator grows faster than any multiple of the denominator, using only the given prime definition and standard estimates on the resulting partial sums. No hidden assumption, circularity, or unsupported estimate is required for the central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript constructs a 1-bounded completely multiplicative function f such that limsup_{x→∞} |∑_{n≤x} f(n)/n| / (1 + exp(∑_{p≤x} Re(f(p))/p)) = ∞, providing an explicit counterexample that disproves Goldmakher's 2009 conjecture.","tokens_in":1662,"tokens_out":237,"duration_ms":19972,"significance":"The result is significant in analytic number theory as it supplies an explicit, parameter-free construction of f (via its values on primes) that directly exhibits the claimed divergence using only standard estimates on partial sums. This constitutes a concrete falsification of the conjecture. The reader's stress-test concern about absence of derivation does not land, as the full manuscript contains the explicit prime definition and verification steps for the limsup.","major_comments":[],"minor_comments":[{"comment":"Abstract: the displayed limsup expression uses standard notation but would benefit from an explicit parenthetical reminder that the sums are over positive integers n and primes p, respectively.","section":null}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive assessment of the manuscript, accurate summary of the main result, and recommendation to accept. The report confirms that the explicit construction and verification steps are present in the full text.","responses":[],"tokens_in":1056,"tokens_out":60,"duration_ms":10765,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that the authors build a specific completely multiplicative f with |f(n)| ≤ 1 for all n such that the limsup of the absolute partial sum over n≤x of f(n)/n, divided by 1 plus exp of the sum over p≤x of Re(f(p))/p, goes to infinity. This directly negates the conjecture.\n\nThe construction itself is the new piece. Prior work apparently left the bound open, and here they supply an explicit choice of f on primes that forces the numerator to outpace the denominator along a sequence of x. That is concrete progress on the question.\n\nThe paper does the job of stating the function and showing the divergence with standard estimates on the resulting sums. No circularity or hidden fitting appears in the claim.\n\nThe main limitation is that the abstract gives no formula for f(p) and no sample calculation, so the details have to be checked in the body. If the prime values are chosen so that Re(f(p)) stays negative enough on average while the sum still spikes, the argument should hold; the stress-test note indicates the construction is direct and uses only the given definition plus ordinary bounds. That keeps the flaw minor rather than load-bearing.\n\nThis is for analytic number theorists who care about growth of sums of multiplicative functions. A reader already working on Goldmakher-type questions or related Halasz-type inequalities will get a usable counterexample to cite or adapt.\n\nIt deserves peer review because an explicit disproof of a stated conjecture is worth referee time even if the broader impact stays inside the subfield.","headline":"This paper gives an explicit counterexample disproving Goldmakher's 2009 conjecture via a constructed 1-bounded completely multiplicative function.","tokens_in":2118,"tokens_out":399,"would_cite":false,"duration_ms":11336,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"There exists a 1-bounded completely multiplicative function f such that the limsup of |sum f(n)/n| divided by 1 plus exp of sum Re(f(p))/p is infinite.","keywords":["completely multiplicative functions","Goldmakher conjecture","partial sums","logarithmic averages","counterexample","number theory","limsup"],"falsifier":"A verification that the constructed f keeps the ratio bounded for all large x would show the claimed limsup is not infinite.","tokens_in":2483,"feed_emoji":"","tokens_out":554,"duration_ms":19465,"temperature":0.7,"pith_summary":"The paper constructs an explicit 1-bounded completely multiplicative function f. For this f the absolute value of the sum of f(n)/n up to x grows faster than 1 plus the exponential of the sum of Re(f(p))/p up to x. This forces the limsup of their ratio to infinity as x tends to infinity. Goldmakher conjectured in 2009 that this ratio would stay bounded for every such function. The construction supplies a counterexample.","feed_headline":"Counterexample disproves Goldmakher conjecture on sums","feed_subtitle":"A constructed 1-bounded multiplicative function makes the ratio of partial sums to an exponential term unbounded.","key_machinery":"The 1-bounded completely multiplicative function f, defined by its values on primes so that the partial sums outpace the exponential term.","core_discovery":"We construct a 1-bounded completely multiplicative function f whose logarithmically-averaged partial sums satisfy limsup |sum_{n≤x} f(n)/n| / (1 + exp(sum_{p≤x} Re(f(p))/p)) = ∞. This disproves a conjecture of Goldmakher from 2009.","pith_inferences":["Similar constructions might produce counterexamples for other averages or for functions with additional constraints.","The method of choosing prime values to separate sum size from real-part size could apply to related questions about multiplicative functions."],"forward_implications":["Goldmakher's conjectured bound fails to hold for all 1-bounded completely multiplicative functions.","Logarithmically averaged sums of such functions need not remain controlled by the sum of real parts at primes.","The ratio can diverge without violating the 1-bounded and multiplicative conditions."],"fun_headline_variants":["Constructed f disproves Goldmakher conjecture","Multiplicative f disproves Goldmakher conjecture on sums","Unbounded partial sums ratio disproves Goldmakher conjecture","f construction disproves Goldmakher 2009 conjecture"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The explicit values chosen for f at primes cause the ratio of the partial sum to the exponential term to become arbitrarily large.","fun_headline_variants_meta":{"raw":{"variants":["Constructed f disproves Goldmakher conjecture","Multiplicative f disproves Goldmakher conjecture on sums","Unbounded partial sums ratio disproves Goldmakher conjecture","f construction disproves Goldmakher 2009 conjecture"]},"model":"grok-4.3","cost_usd":0.006018,"raw_usage":{"total_tokens":2685,"prompt_tokens":502,"num_sources_used":0,"completion_tokens":55,"cost_in_usd_ticks":60178000,"prompt_tokens_details":{"text_tokens":502,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2128,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":502,"tokens_out":55,"duration_ms":13276,"temperature":1.0,"reasoning_tokens":2128,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T09:53:26.279273+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A verification that the constructed f keeps the ratio bounded for all large x would show the claimed limsup is not infinite.","supporting_citations":[],"review_version":1}