{"id":"bf755c4c-14ff-4662-853d-2bbf0669d9c3","arxiv_id":"2607.04366","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The additive Mertens sum Sk(x) admits a complete asymptotic series in powers of 1/log x whose coefficients are explicit multiple logarithmic integrals, with closed forms for low orders and all orders when k=2.","lead":"The paper gives a complete asymptotic expansion for the sum of 1 over sums of k primes each at most x. This fills in the additive counterpart to classical Mertens theorems and supplies explicit coefficients for small k.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The paper’s strongest claim is a complete asymptotic expansion whose coefficients are explicit multiple integrals. The only non-elementary input is the classical PNT error term used in Lemma 2; every subsequent step (rescaling, Taylor remainder, absolute convergence, closed-form extraction via generating functions and finite differences) is elementary real analysis and is written out in detail. The numerical tables for k=2 and k=3 already give independent corroboration of the first few coefficients. Because the classical error is far stronger than needed, the reader’s identified weakest assumption does not threaten the claim under any presently known form of the prime-number theorem. Consequently the ACCEPT verdict with high confidence stands; no adjustment is required.","tokens_in":19001,"tokens_out":482,"duration_ms":5219,"concrete_test":"Independently recompute the first three coefficients E2,0, E2,1, E2,2 from the closed formulae of Theorems 2–3 and compare them with high-precision numerical quadrature of the defining integrals (2) over (0,1]2; agreement to ≥10-12 digits would reconfirm both the Laplace-reduction machinery and the extraction of Δk-1Hk(0).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 1) rests on writing Sk as a Stieltjes integral, expanding the product of dπ = du/log u + dR, isolating the pure main term Mk, and showing every term with at least one dR is O(xk-1 exp(-c'√log x)) (Lemma 2). The classical zero-free-region error for R is more than enough for that bound, and the subsequent rescaling + multivariate Taylor argument that produces the Ek,n series is standard and carefully controlled (absolute convergence of the integrals is proved in Lemma 1). The reader correctly flags the PNT error as the weakest link, but under any error term currently known for π(x) the complete expansion to arbitrary N remains valid. No internal inconsistency or hidden assumption that would invalidate the expansion was found.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper introduces the additive Mertens sum Sk(x)=∑_{p1,...,pk≤x} 1/(p1+⋯+pk) for fixed k≥2 and proves a complete asymptotic expansion Sk(x)=(x^{k-1}/log^k x)∑_{n=0}^N Ek,n/log^n x + O(x^{k-1}/log^{k+N+1}x) for every N≥0. The coefficients are the absolutely convergent integrals Ek,n=(-1)^n ∫_{(0,1]^k} hn(log t)/(∑t) dt with hn the complete homogeneous symmetric polynomial of degree n. Closed forms are obtained for Ek,0 and Ek,1 for all k, for the diagonal part of Ek,2 for all k (and fully for Ek,2 when k≤4), and for the entire sequence {E2,n} as explicit Q-linear combinations of log 2 and zeta values. The argument proceeds from the Stieltjes representation of Sk, the classical PNT error, isolation of the pure main term Mk, rescaling, and multivariate Taylor expansion with controlled remainder; numerical checks for k=2,3 corroborate the first few terms.","tokens_in":19183,"tokens_out":760,"duration_ms":7018,"significance":"The additive Mertens sum appears not to have been studied previously; the paper supplies a complete asymptotic expansion of arbitrary order together with explicit arithmetic expressions for the first coefficients and a closed formula for the full sequence when k=2. The derivation is elementary (real-variable PNT, Laplace transforms, finite differences, Taylor remainder) and free of fitted parameters. The numerical tables for S2 and S3 provide independent verification of the closed forms. The work therefore fills a natural gap left by the multiplicative theory of Tenenbaum, Popa, Qi–Hu and supplies concrete, computable expansions that can be used in further analytic-number-theoretic estimates.","major_comments":[],"minor_comments":[{"comment":"In the abstract and Theorem 1 the O-term is written without the subscript k,N that appears later; a uniform notation would avoid any ambiguity about dependence of the implied constant.","section":null},{"comment":"Section 4.3 and the proof of Theorem 2(ii) invoke symbolic verification for k=2….7; a short remark on the computer-algebra system used (or a reference to a supplementary notebook) would strengthen reproducibility.","section":null},{"comment":"Proposition 2 for B4 lists a lengthy expression inside the Q-algebra generated by L3; a brief note that the representation is not unique (already mentioned in a footnote) could be moved into the main text for clarity.","section":null},{"comment":"Table 1 and the numerical Tables 2–3 are clear, but the caption of Figure 1 could explicitly state that the curves are relative errors |approx-exact|/exact rather than absolute errors.","section":null},{"comment":"A few minor typographical inconsistencies appear (e.g., “coeüients” vs. “coefficients”, occasional missing spaces around “=”); a light copy-edit would remove them.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is solid and self-contained. The only potential scope question is whether a short note on an additive analogue is of sufficient interest for the journal; given the complete expansion and the closed forms for k=2, I believe it is. No citation or novelty concerns."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper introduces the additive Mertens sum Sk(x) = sum 1/(p1+…+pk) over primes ≤ x and proves a complete asymptotic expansion to arbitrary order, with coefficients given by explicit multiple integrals involving complete homogeneous symmetric polynomials. That object appears to be new; the literature they cite (Tenenbaum, Popa, Qi–Hu) treats only the multiplicative versions.\n\nWhat they do well is the real-variable analysis. They write Sk as a Stieltjes integral, expand dπ = du/log u + dR, show every term with a dR factor is O(x^{k-1} exp(-c'√ log x)) by integration by parts (Lemma 2), rescale the main term, and extract the series via multivariate Taylor remainder with an absolute-convergence proof for the coefficient integrals (Lemmas 1 and 3). The classical PNT error is more than enough for the complete expansion; under any currently known error term the claim remains valid. Closed forms for Ek,0 and Ek,1 for all k, the diagonal part of Ek,2, full explicitness for k≤4, and a clean eta/zeta formula for the entire sequence when k=2 are genuine additions. Numerical checks for S2 and S3 match the first few terms and illustrate the asymptotic (non-convergent) character of the series.\n\nSoft spots are minor and already flagged by the authors. The cross terms in Ek,2 for k≥5 introduce higher polylogarithms that do not reduce to a simple basis; they leave this as an open structural obstacle rather than claiming a unified closed form. The expansion is asymptotic, not convergent, which is standard. Citation pattern is appropriate and focused.\n\nThis is for people who work on prime sums, Mertens-type theorems, or explicit asymptotics in analytic number theory. The central argument is sound, the derivations are transparent, and there is no circularity or free parameters. I would send it to a serious referee without hesitation; it is a clean, self-contained contribution that organises a previously unstudied family of sums.","headline":"Solid, carefully executed complete asymptotic expansion for a new additive Mertens sum; the math holds and the limitations are stated honestly.","tokens_in":19784,"tokens_out":529,"would_cite":true,"duration_ms":5437,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N05","11M06","33B30","41A60"],"pacs":[],"model":"grok-4.5","headline":"The additive sum of reciprocal prime sums admits a complete asymptotic expansion whose coefficients are multiple logarithmic integrals over the unit cube.","keywords":["additive Mertens sum","asymptotic expansion","complete homogeneous symmetric polynomial","multiple logarithmic integrals","prime number theorem","Dirichlet eta function","polylogarithms"],"falsifier":"Compute the exact double or triple sum Sk(x) by prime convolution for several large x (e.g., 10^6–10^7) and check whether the successive partial sums of the claimed series reduce the relative error by roughly a factor of three each time, matching the explicit numerical tables already given for k=2 and k=3.","tokens_in":19913,"feed_emoji":"∑","tokens_out":670,"duration_ms":5601,"temperature":0.7,"pith_summary":"Classical Mertens theorems describe sums of reciprocal primes; their multiplicative generalisations grow like powers of log log x. This paper studies the additive counterpart: the sum of 1/(p1+…+pk) over all primes ≤x. That sum grows like x^{k-1}/(log x)^k. The authors prove that it possesses a full asymptotic series in descending powers of log x, with every coefficient given by an absolutely convergent integral of a complete homogeneous symmetric polynomial against 1/(t1+…+tk) on the unit cube. They evaluate the first two coefficients in closed form for every k, the third coefficient completely for k≤4, and the entire infinite sequence of coefficients when k=2 (as explicit rational combinations of log 2 and zeta values). The result therefore supplies precise leading constants and arbitrarily many secondary terms for an elementary but previously unstudied prime sum.","feed_headline":"Additive prime sum gets full asymptotic series","feed_subtitle":"Coefficients are explicit cube integrals; first three terms closed for k=2,3","key_machinery":"The coefficient integrals Ek,n themselves: after the prime-number theorem converts the original Stieltjes integral into a smooth main term, a linear rescaling uj=xtj and a multivariate Taylor expansion of the product of (1+wj)^{-1} factors produce exactly these integrals as the successive coefficients.","core_discovery":"For every fixed k≥2 and every N≥0 the additive Mertens sum admits the complete expansion Sk(x)=(x^{k-1}/log^k x)∑_{n=0}^N Ek,n/log^n x + O(x^{k-1}/log^{k+N+1}x), where the coefficients are the absolutely convergent integrals Ek,n=(-1)^n ∫_{(0,1]^k} hn(log t1,…,log tk)/(t1+…+tk) dt and hn is the complete homogeneous symmetric polynomial of degree n.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Additive Mertens sum Sk(x) admits full asymptotic series","Complete log-power expansion for sums over k prime reciprocals","Sk(x) expands as x^{k-1}/log^k x times 1/log series","Explicit multi-integrals give all coeffs of additive Mertens asymptotics","Full series for additive prime-sum Mertens via homogeneous polynomials"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The classical prime-number theorem error is small enough that every term containing at least one residual factor is smaller than any negative power of log x; a weaker error would stop the expansion at finite order.","fun_headline_variants_meta":{"raw":{"variants":["Additive Mertens sum Sk(x) admits full asymptotic series","Complete log-power expansion for sums over k prime reciprocals","Sk(x) expands as x^{k-1}/log^k x times 1/log series","Explicit multi-integrals give all coeffs of additive Mertens asymptotics","Full series for additive prime-sum Mertens via homogeneous polynomials"]},"model":"grok-4.5","effort":"low","cost_usd":0.0071,"raw_usage":{"total_tokens":1966,"prompt_tokens":1078,"num_sources_used":0,"completion_tokens":84,"cost_in_usd_ticks":71000000,"prompt_tokens_details":{"text_tokens":1078,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":804,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":1078,"tokens_out":84,"duration_ms":6445,"temperature":1.0,"reasoning_tokens":804,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-13T06:59:20.134975+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Compute the exact double or triple sum Sk(x) by prime convolution for several large x (e.g., 10^6–10^7) and check whether the successive partial sums of the claimed series reduce the relative error by roughly a factor of three each time, matching the explicit numerical tables already given for k=2 and k=3.","supporting_citations":[],"review_version":2}