{"id":"c12690d4-ee04-4c3e-a8cd-5c71662f1577","arxiv_id":"2606.29930","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Extends variance formulas for digits of 1/p in base b to period lengths (p-1)/2^m using Dedekind sums, class numbers, and generalized Bernoulli numbers.","lead":"The paper extends prior formulas for the variance of digits in the base-b expansion of 1/p to the case where the period length l equals (p-1) divided by a power of 2. A smart generalist might read it for insight into how number-theoretic invariants describe statistical features of repeating digit sequences.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's UNVERDICTED status stems directly from the absence of the manuscript derivations. Because the present review likewise lacks any concrete mathematical steps to scrutinize, the same verdict is retained; no new load-bearing concern is generated.","tokens_in":1673,"tokens_out":274,"duration_ms":19999,"concrete_test":"For the smallest prime p>3 with ord_p(b)=(p-1)/4 (e.g., p=13, b=2), compute the digit variance of one full period of 1/p in base b by direct summation; compare the numerical value against the closed-form expression claimed in the paper for m=2; agreement to machine precision would confirm the formula holds at least in this case.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract states that closed-form expressions for the variance are obtained in terms of Dedekind sums, class numbers, and generalized Bernoulli numbers when the period l equals (p-1)/2^m for m≥1. No derivation, equation, or intermediate step is supplied in the query, so no specific assumption (e.g., an implicit bound on a sum, a hidden identity, or a convergence issue for m>1) can be isolated as internally inconsistent or unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper extends prior results on the mean and variance of digits in the base-b expansion of 1/p (p prime, b not divisible by p) to the case where the multiplicative order l of b modulo p equals (p-1)/2^m for integers m ≥ 1. It derives closed-form expressions for these quantities in terms of Dedekind sums, class numbers of imaginary quadratic fields, and generalized Bernoulli numbers, thereby covering and generalizing the previously treated cases l = p-1 and l = (p-1)/2.","tokens_in":1736,"tokens_out":324,"duration_ms":31203,"significance":"If the derivations are correct, the work supplies a uniform number-theoretic framework for digit statistics over an infinite family of periods indexed by powers of 2. This strengthens the connection between periodic digit expansions and classical objects (Dedekind sums, class numbers, Bernoulli numbers) and may enable explicit computations or further arithmetic applications for these special periods.","major_comments":[],"minor_comments":[{"comment":"The abstract states that formulas 'were given previously' for l = p-1 and l = (p-1)/2 but does not cite the specific references; adding these citations would improve context.","section":"Abstract"},{"comment":"Notation for the generalized Bernoulli numbers and the precise range of m should be introduced explicitly in the introduction to avoid any ambiguity for readers unfamiliar with the prior literature.","section":"Introduction"}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive report and recommendation to accept the manuscript. We are pleased that the work is viewed as providing a uniform framework connecting digit statistics to classical arithmetic objects.","responses":[],"tokens_in":1161,"tokens_out":55,"duration_ms":12128,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that Girstmair has carried the earlier variance calculations for the digits of 1/p in base b over to the periods l = (p-1)/2^m for all m ≥ 1. The abstract makes clear that the same objects—Dedekind sums, class numbers of imaginary quadratic fields, and generalized Bernoulli numbers—still deliver closed forms.\n\nThe work does what it sets out to do: it treats the general m case as a single theory rather than a collection of special cases. The fact that it recovers the m=1 situation as a check is useful. The approach stays inside the existing number-theoretic toolkit, so the extension looks mechanical but not trivial.\n\nThe obvious soft spot is that only the abstract is in front of us. Without the actual derivations it is impossible to see how the sums and class-number terms behave when m increases, or whether extra correction terms appear that were absent for m=0,1. That leaves the soundness claim provisional.\n\nThe paper is aimed at the small group of people already tracking explicit formulas for digit statistics in rational base-b expansions. A reader who knows the two earlier papers will see immediately what has been added; outsiders will not find much to use.\n\nThe central claim is narrow but well-defined and appears free of circularity or post-hoc fitting. I would send it to referees rather than desk-reject it, on the grounds that the program is coherent and the new formulas are stated explicitly enough to be checked.","headline":"This extends the known variance formulas for digits in 1/p expansions from the full and half-period cases to the full family of periods (p-1)/2^m.","tokens_in":2265,"tokens_out":388,"would_cite":false,"duration_ms":26224,"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":"The variance of digits in the base-b expansion of 1/p admits closed-form expressions in Dedekind sums, class numbers, and generalized Bernoulli numbers when the period length equals (p-1)/2^m for m at least 1.","keywords":["mean values","variances","digits of 1/p","periodic expansions","Dedekind sums","class numbers","generalized Bernoulli numbers","multiplicative order"],"falsifier":"Pick a prime p such as 41 where (p-1)/4 = 10, fix base b=10, compute the exact variance of the ten digits in one period of 1/41 by direct addition, and compare the numerical value against the closed-form prediction from the Dedekind-sum expression.","tokens_in":2540,"feed_emoji":"","tokens_out":720,"duration_ms":28977,"temperature":0.7,"pith_summary":"The paper extends earlier formulas for the mean and variance of digits in the repeating expansion of 1/p. It covers the case where the multiplicative order of b modulo p is exactly (p-1) divided by a power of 2. The resulting expressions rely on the same arithmetic objects that appeared in the full-period and half-period cases. A reader would care because the formulas turn long digit sums into exact evaluations that depend only on invariants of p.","feed_headline":"Closed-form variance for digits of 1/p when period is (p-1)/2^m","feed_subtitle":"Formulas in Dedekind sums and class numbers now cover all cases where the order divides (p-1) by a power of two.","key_machinery":"Closed-form expressions for the variance of the digit sum over a full period, obtained by reducing the sum via the order l = (p-1)/2^m to combinations of Dedekind sums and class-number terms.","core_discovery":"For a prime p not dividing b, when the period length l equals (p-1)/2^m with m at least 1, both the mean value and the variance of the digits in one full period of the base-b expansion of 1/p are given by explicit formulas involving Dedekind sums, class numbers of imaginary quadratic fields, and generalized Bernoulli numbers.","pith_inferences":["The method may extend to other proper divisors of p-1 that are not powers of two.","Closed forms of this type could be used to test conjectures on the statistical uniformity of digits for primes with restricted orders.","Links between digit variances and class numbers might produce new relations between quadratic fields and base-b expansions."],"forward_implications":["The earlier results for full period l = p-1 and for l = (p-1)/2 become immediate special cases of the new theory.","Exact variances become computable for any prime whose order is (p-1) divided by a power of two, without enumerating the digits.","The same arithmetic invariants control both the mean and the variance across this family of periods.","The approach supplies a uniform method that works uniformly for all m at least 1."],"fun_headline_variants":["Mean and variance of 1/p digits for periods (p-1)/2^m","Dedekind sum formulas for digit variance in 1/p with order (p-1)/2^m","Class numbers determine 1/p digit variances when l=(p-1)/2^m","Bernoulli numbers in explicit 1/p digit mean and variance formulas"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The variance of the digits admits closed-form expressions in terms of Dedekind sums, class numbers of imaginary quadratic fields, and generalized Bernoulli numbers when the multiplicative order l equals (p-1)/2^m.","fun_headline_variants_meta":{"raw":{"variants":["Mean and variance of 1/p digits for periods (p-1)/2^m","Dedekind sum formulas for digit variance in 1/p with order (p-1)/2^m","Class numbers determine 1/p digit variances when l=(p-1)/2^m","Bernoulli numbers in explicit 1/p digit mean and variance formulas"]},"model":"grok-4.3","cost_usd":0.003455,"raw_usage":{"total_tokens":1710,"prompt_tokens":602,"num_sources_used":0,"completion_tokens":89,"cost_in_usd_ticks":34553000,"prompt_tokens_details":{"text_tokens":602,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1019,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":602,"tokens_out":89,"duration_ms":8109,"temperature":1.0,"reasoning_tokens":1019,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-30T05:29:57.420270+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Pick a prime p such as 41 where (p-1)/4 = 10, fix base b=10, compute the exact variance of the ten digits in one period of 1/41 by direct addition, and compare the numerical value against the closed-form prediction from the Dedekind-sum expression.","supporting_citations":[],"review_version":1}