{"id":"e95945d1-7b27-454f-b79c-5f51ca858d5c","arxiv_id":"2607.01956","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":4.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Explicit asymptotic formula proved for smoothed sum of σ(n), yielding convergence of integral for Walfisz error term.","lead":"The authors prove an explicit asymptotic formula for the smoothed sum of the sum-of-divisors function σ(n) weighted by 1-x/n. This removes a difficult term from the average order of σ(n) and implies convergence of an integral tied to the error term in the Walfisz divisor problem.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Accuracy of explicit constants from MVT/Euler-Maclaurin in appendix for error control","rationale":"The reader's weakest_assumption correctly isolates the single point at which the argument is most sensitive; without independent verification of those constants the explicit claims remain formally unconfirmed, which is why UNVERDICTED with low confidence is the appropriate stance.","tokens_in":1625,"tokens_out":277,"duration_ms":12466,"concrete_test":"Re-derive the two or three leading explicit constants in the appendix from first principles (MVT on the relevant interval plus Euler-Maclaurin with the same number of terms), then substitute them back into the error estimate for the smoothed sum; if the resulting bound exceeds the main term by more than a factor of 2, the asymptotic and integral convergence statements do not follow.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that the explicit bounds obtained in the appendix via the mean value theorem and Euler-Maclaurin formula are sufficiently sharp to (i) absorb the difficult remainder into the main asymptotic for the weighted sum of σ(n) and (ii) guarantee absolute convergence of the integral involving the Walfisz error term. If any of those derived constants is understated by even a modest factor, both the elimination step and the corollary fail.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper proves a totally explicit asymptotic formula for the sum of σ(n) twisted by the weight 1-x/n. This is used to eliminate the difficult remainder term in the classical average order of σ(n). As a corollary, the authors deduce the convergence of an integral involving the error term from the Walfisz divisor problem. The proofs rely on an appendix supplying explicit estimates obtained via the mean value theorem and the Euler-Maclaurin summation formula.","tokens_in":1698,"tokens_out":529,"duration_ms":17178,"significance":"If the explicit constants derived in the appendix are sufficiently sharp, the result supplies a concrete, parameter-free handle on a smoothed sum-of-divisors sum and resolves a convergence question tied to the Walfisz error term. Such explicit control is valuable in analytic number theory when one wishes to pass from smoothed to unsmoothed statements or to justify integral representations of remainder terms.","major_comments":[{"comment":"Appendix (explicit bounds via MVT and Euler-Maclaurin): the manuscript asserts that the derived constants suffice both to absorb the difficult remainder into the main asymptotic for the weighted sum of σ(n) and to guarantee absolute convergence of the integral in the Walfisz corollary. No numerical check or comparison with known sharper bounds is supplied; if any constant is understated by a modest factor, both the elimination step and the corollary fail. A concrete verification (e.g., explicit numerical evaluation of the leading error term for a moderate x) is required.","section":"Appendix"},{"comment":"Main theorem (asymptotic for the weighted sum): the claim that the explicit formula eliminates the difficult part of the classical average order of σ(n) rests entirely on the error term being smaller than the main term after the constants from the appendix are inserted. The manuscript does not display the numerical size of this error term relative to the main term for any concrete range of x, leaving the load-bearing comparison unverified.","section":"Main theorem"}],"minor_comments":[{"comment":"Notation for the weight function 1-x/n should be introduced with a displayed equation and a clear statement of the range of summation.","section":"Introduction"},{"comment":"The abstract and introduction both refer to “the difficult part in the classical average order of σ(n)” without a precise citation to the classical formula or the precise remainder that is being removed.","section":"Introduction"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and for highlighting the need for explicit verification of the bounds. We respond point by point to the major comments.","responses":[{"response":"The constants are obtained by applying the mean value theorem and Euler-Maclaurin formula while retaining explicit remainder terms at every step; the subsequent proofs then insert these constants directly into the estimates for the weighted sum and the Walfisz integral, showing analytically that the required inequalities hold. We acknowledge that an independent numerical check would increase , and we will add such a verification (evaluation of the leading error term at x=10^4 together with the bound) to the revised appendix.","revision_made":"partial","referee_comment":"[Appendix] Appendix (explicit bounds via MVT and Euler-Maclaurin): the manuscript asserts that the derived constants suffice both to absorb the difficult remainder into the main asymptotic for the weighted sum of σ(n) and to guarantee absolute convergence of the integral in the Walfisz corollary. No numerical check or comparison with known sharper bounds is supplied; if any constant is understated by a modest factor, both the elimination step and the corollary fail. A concrete verification (e.g., explicit numerical evaluation of the leading error term for a moderate x) is required."},{"response":"The explicit formula supplies a concrete error term whose size relative to the main term is controlled by the appendix constants; the absorption is therefore a direct (if tedious) consequence of those inequalities. To make the comparison transparent we will insert, in the revised main theorem section, a short table or paragraph displaying the numerical ratio of error to main term for several moderate values of x (e.g., 10^3 to 10^5).","revision_made":"partial","referee_comment":"[Main theorem] Main theorem (asymptotic for the weighted sum): the claim that the explicit formula eliminates the difficult part of the classical average order of σ(n) rests entirely on the error term being smaller than the main term after the constants from the appendix are inserted. The manuscript does not display the numerical size of this error term relative to the main term for any concrete range of x, leaving the load-bearing comparison unverified."}],"tokens_in":1294,"tokens_out":480,"duration_ms":19413,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper extends the authors' earlier smoothed Dirichlet-divisor work from τ(n) to σ(n). They give a fully explicit asymptotic for the weighted sum of σ(n) with weight 1 - x/n, which removes the hard remainder from the usual average order, and they deduce convergence of an integral over the Walfisz error term.\n\nThe concrete formulas and the appendix with mean-value and Euler-Maclaurin bounds are the useful parts. Specialists who need explicit constants rather than asymptotic statements can plug these in directly.\n\nThe soft spot is exactly the one flagged in the stress test: whether the explicit constants coming out of the appendix are sharp enough to absorb the remainder and guarantee absolute convergence of the integral. If any bound is understated by even a modest factor, both the main elimination step and the corollary fail. The methods are standard, so the issue is accuracy of the derived numbers, not the overall strategy.\n\nThis is for people already working on explicit estimates in divisor problems. A reader in that narrow corner will get usable formulas and a checkable corollary. It is worth sending to peer review because the claims are concrete enough to verify or correct.","headline":"This paper gives an explicit smoothed asymptotic for the weighted sum of σ(n) plus a convergence corollary for the Walfisz integral, extending the authors' prior τ work, but the appendix constants are the part that needs checking.","tokens_in":2154,"tokens_out":326,"would_cite":false,"duration_ms":15543,"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":"A fully explicit asymptotic for the weighted sum of σ(n) removes the main error term in its average order.","keywords":["sum-of-divisors function","smoothed sums","Walfisz divisor problem","explicit asymptotics","Euler-Maclaurin formula","mean value theorem","error terms"],"falsifier":"Numerical evaluation of the integral over successively larger intervals to test whether the partial integrals remain bounded, or direct comparison of the stated asymptotic against computed values of the weighted sum for large explicit x.","tokens_in":2521,"feed_emoji":"","tokens_out":605,"duration_ms":18589,"temperature":0.7,"pith_summary":"The paper establishes a completely explicit asymptotic formula for the sum of σ(n) twisted by the smoothing weight 1 - x/n. This formula lets the authors bypass the difficult error term that normally appears when estimating the average order of the sum-of-divisors function. As a corollary they obtain the convergence of a certain integral that encodes the error term in the Walfisz divisor problem. The derivation rests on explicit bounds supplied in an appendix that applies the mean value theorem and the Euler-Maclaurin summation formula.","feed_headline":"Explicit asymptotic for weighted σ(n) sum proves integral convergence","feed_subtitle":"The formula eliminates the hard error term in the average order of the sum-of-divisors function.","key_machinery":"The explicit asymptotic formula for the sum of σ(n) weighted by 1-x/n, obtained from the mean value theorem and Euler-Maclaurin summation.","core_discovery":"We prove a totally explicit asymptotic formula for the sum of σ(n) twisted by the weight 1-x/n, which enables us to eliminate the difficult part in the classical average order of σ(n). As a corollary, we deduce the convergence of an integral dealing with the error term in the Walfisz divisor problem.","pith_inferences":["The same smoothing technique might be applied to obtain explicit formulas for other divisor functions such as τ(n).","Convergence of the integral implies improved integrability properties for the error term in related divisor problems.","The method could be tested on higher-order moments of σ(n) to check whether similar explicit asymptotics hold."],"forward_implications":["The difficult part in the classical average order of σ(n) is eliminated by the explicit formula.","Convergence is established for the integral that controls the error term in the Walfisz divisor problem.","The appendix supplies explicit constants that support both the main formula and the convergence claim."],"fun_headline_variants":["Weighted σ(n) asymptotic proves integral convergence","Twisted σ(n) sum yields Walfisz integral convergence","Asymptotic eliminates difficult part in σ(n) average order","Explicit formula for weighted σ(n) sum deduces convergence"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The explicit constants obtained from the mean value theorem and Euler-Maclaurin summation formula in the appendix are sufficiently accurate to justify both the main asymptotic formula and the convergence of the integral.","fun_headline_variants_meta":{"raw":{"variants":["Weighted σ(n) asymptotic proves integral convergence","Twisted σ(n) sum yields Walfisz integral convergence","Asymptotic eliminates difficult part in σ(n) average order","Explicit formula for weighted σ(n) sum deduces convergence"]},"model":"grok-4.3","cost_usd":0.009711,"raw_usage":{"total_tokens":4260,"prompt_tokens":536,"num_sources_used":0,"completion_tokens":63,"cost_in_usd_ticks":97112000,"prompt_tokens_details":{"text_tokens":536,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3661,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":536,"tokens_out":63,"duration_ms":23354,"temperature":1.0,"reasoning_tokens":3661,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-03T07:05:45.484277+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Numerical evaluation of the integral over successively larger intervals to test whether the partial integrals remain bounded, or direct comparison of the stated asymptotic against computed values of the weighted sum for large explicit x.","supporting_citations":[],"review_version":1}