{"id":"f8c2b064-69e2-47fa-bf10-25e34a1c8464","arxiv_id":"2607.08985","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For every fixed k≥4 the kth moment of r2(n) satisfies ∑_{n≤x} r2(n)^k ≃_k x (log x)^{2^{k-1}-2k-1}.","lead":"The paper proves matching upper and lower bounds of the correct order for the kth moments of the number of representations of n as a sum of two prime squares, for every fixed k≥4. This removes a previous logloglog factor in the k=4 upper bound and makes the lower bounds unconditional, with a simpler proof of related lower bounds for the shifted-prime divisor function.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the geometric-mean step in Prop. 5.1 as the only non-routine technical point, yet that step is standard for upper bounds and the saving exponent 1/(32K) supplied by Sabuncu’s Lemma 5.6 is a fixed positive power for each fixed k. Combined with the rapid decay of the Dickman function, it removes the log-log-log factor without requiring any new conjecture. The lower-bound argument via mixed moments is unconditional and elementary once Bombieri–Vinogradov is granted. Consequently the manuscript supplies precisely the missing unconditional order-of-magnitude statements, and the ACCEPT verdict with high confidence stands.","tokens_in":17942,"tokens_out":516,"duration_ms":5377,"concrete_test":"Independently recompute the Euler product that appears after the geometric mean in the proof of Prop. 5.1 (the sum over square-free ℓ2 of μ^{2}(ℓ2)(4k)^{ω(ℓ2)} τ_K(ℓ2)^{1/2} / ℓ2^{1+1/(64K)}) for k=4 and verify absolute convergence; then check that the resulting Φ4(u) still yields ∑_ℓ ℓ^{2^{3}-2·4-2} Φ4(u_ℓ) ≪ (log x)^{2^{3}-2·4-1} after the e-adic summation of §6.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 1.1 / Prop. 2.1) rests on two independent pillars: (i) the mixed-moment lower bound Mk(x) via Hölder + Bombieri–Vinogradov averages (Lemmas 3.1–3.3, §7), and (ii) the medium-range singular-series average (Prop. 5.1) that inserts a square-root Dickman factor into Sabuncu’s e-adic sum. Both are executed inside the classical ranges of the cited theorems (Tenenbaum–Wu friable Selberg–Delange, Bombieri–Vinogradov, Selberg sieve). The geometric-mean step that produces ρ^{1/2} is legitimate for upper bounds and the resulting decay is more than enough to absorb the log-log-log that previously appeared for k=4. No hidden uniformity failure, circularity, or range violation is visible.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper establishes the correct order of magnitude for the kth moments of r2(n), the number of representations of n as a sum of two prime squares, for every fixed integer k≥4: ∑_{n≤x} r2(n)^k ≃_k x (log x)^{2^{k-1}-2k-1}. After the elementary reduction (2.1) to the ordered counting function R2, the upper bound is obtained from factorial moments Sk(x) (Proposition 2.1) via Sabuncu’s Gaussian factorization and Selberg-sieve framework, with a new friable Selberg–Delange input (Tenenbaum–Wu) that inserts a Dickman-type factor into the medium-prime range and removes the previous log log log x loss for k=4. The matching lower bound is obtained unconditionally by replacing the pure moment of R2 by the mixed moment Mk(x)=∑ R2(n) r0(n)^{k-1}, applying Hölder, and estimating the mixed moment via the character expansion of r0 together with Bombieri–Vinogradov averages for prime pairs (Lemmas 3.1–3.3, §7). As a corollary one obtains the optimal error term for the third moment of r2. The same mixed-moment method yields a simpler unconditional proof of the lower bounds for the moments of the shifted-prime divisor function ω*, recovering the lower-bound half of Gabdullin’s resolution of the Fan–Pomerance conjecture.","tokens_in":18152,"tokens_out":1182,"duration_ms":9905,"significance":"The result closes the remaining gap left by Sabuncu for the moments of r2: the upper bound for k=4 is now of the expected order, and all lower bounds for k≥4 are unconditional. The mixed-moment reduction (keeping one prime-square representation and supplying the rest from r0) is a clean, reusable idea that avoids any appeal to a uniform Green–Tao theorem and immediately extends to ω* and, as the author indicates, to r1. The upper-bound improvement is likewise concrete: the friable average of the singular series (Proposition 5.1) produces a square-root Dickman factor that is more than enough to absorb the previous logarithmic loss. Both pillars rest on classical tools (Bombieri–Vinogradov, Selberg sieve, Tenenbaum–Wu) applied inside their standard ranges, so the paper supplies a definitive unconditional statement of the expected order of magnitude for these moments.","major_comments":[],"minor_comments":[{"comment":"In the abstract and introduction the arXiv identifier of Sabuncu is written [Sabuncu2024] while the bibliography uses [Sab24]; unify the citation key.","section":"Abstract / Bibliography"},{"comment":"Section 2.1, display after (2.2): the exponent arithmetic that converts Mk(x) ≫ x (log x)^{2^{k-1}-3} into the pure-moment lower bound is correct but written in a single dense line; a short intermediate step would help the reader.","section":"Section 2.1"},{"comment":"Lemma 3.2: the application of Cauchy–Schwarz followed by the trivial bound E(m) ≪ P log log(m+2)/m is standard, but the final choice of A relative to B is left implicit; a one-line remark that A can be taken larger than any fixed power of B would make the dependence transparent.","section":"Lemma 3.2"},{"comment":"Proposition 5.1: the geometric-mean step that produces \rho_{2^{k-1}}(u)^{1/2} is legitimate for upper bounds, yet a brief sentence noting that any positive power of the divisor saving would suffice would clarify the robustness of the argument.","section":"Proposition 5.1"},{"comment":"Section 8, display (8.6): the local factor g_k(ℓ) is written correctly, but the subsequent appeal to Selberg–Delange for ∑ g_k(c)/φ(c) would benefit from an explicit reference to the same argument used in Lemma 3.3.","section":"Section 8"},{"comment":"A few typographical inconsistencies appear (e.g., “Erdös” vs. “Erdős”, occasional missing spaces around ≃ and ≪). A light copy-edit pass would remove them.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is a clean, self-contained improvement of Sabuncu’s work that removes both the conditional lower bounds and the residual log-log-log factor. It is well within the scope of a strong analytic-number-theory journal and requires no further technical revision. The mixed-moment idea is likely to be cited independently of the r2 application."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The punchline is simple: Gou finishes the moment problem for r_{2} that Sabuncu left open. For every fixed k≥4 one now has the correct order ∑_{n≤x} r_{2}(n)^k ≃_k x (log x)^{2^{k-1}-2k-1}, unconditionally. The k=4 upper bound loses its extra logloglog factor, the third-moment error reaches the O(x/(log x)^3) that Blomer–Brüdern expected, and the same mixed-moment idea gives a shorter proof of the lower bounds for ω*.\n\nWhat is new is the lower-bound device. Instead of counting pure k-tuples of prime-square representations (which forced Sabuncu to invoke a uniform Green–Tao conjecture), Gou keeps one genuine R_{2} factor and supplies the remaining multiplicity by r_{0}^{k-1}. Hölder plus the known moments of r_{0} reduce everything to a mixed moment that is accessible by Bombieri–Vinogradov averages over m | p^{2}+q^{2} with m square-free and primes ≡1 mod 4. That is clean classical analytic number theory; Lemmas 3.1–3.3 and Section 7 are written carefully and stay inside the known ranges.\n\nOn the upper side the skeleton is Sabuncu’s (Gaussian factorisation, Selberg sieve, largest-prime-factor split). The only genuine improvement is the insertion of the Tenenbaum–Wu friable Selberg–Delange estimate into the medium range. Taking the geometric mean of a pure friable r_{0}^k bound and Sabuncu’s divisor-saving estimate produces a square-root Dickman factor that is more than enough to absorb the harmonic sum that previously produced logloglog for k=4. The resulting Proposition 5.1 and the e-adic summation in Section 6 look correct.\n\nSoft spots are minor. The geometric-mean step is a bit crude, but legitimate for upper bounds and not load-bearing. The ω* application is only a sketch, yet it recovers the lower bound of Gabdullin with less combinatorial overhead. Citation pattern is honest: Sabuncu, Blomer–Brüdern, Tenenbaum–Wu, and the classical tools are all properly placed.\n\nThis is a solid specialist paper for people who already care about moments of representation functions by primes. It deserves a serious referee and should be accepted after the usual polishing. I would cite the mixed-moment idea and the clean third-moment error term.","headline":"Unconditional order-of-magnitude for all higher moments of r₂, via a clean mixed-moment lower bound and a friable refinement that kills the last logloglog.","tokens_in":18772,"tokens_out":682,"would_cite":true,"duration_ms":8011,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N37","11P32","11A25"],"pacs":[],"model":"grok-4.5","headline":"The kth moments of representations as sums of two prime squares have the expected size for every fixed k≥4, unconditionally.","keywords":["prime squares","representation functions","moments","Selberg sieve","friable integers","shifted prime divisor function"],"falsifier":"If an explicit computation of the fourth moment up to a large x (say 10^{12}) produced a growth rate visibly larger than x/(log x)^5 by more than a slowly growing log-log-log factor, the claimed upper bound would be false.","tokens_in":18835,"feed_emoji":"∑","tokens_out":675,"duration_ms":8098,"temperature":0.7,"pith_summary":"How often can an integer be written as a sum of two prime squares, and how large do those representation counts get on average? This paper shows that the kth power of the representation function r2, summed up to x, has the correct order of magnitude x times a precise power of log x, for every fixed k at least 4. The matching lower bounds no longer need a conjectural uniform Green–Tao statement about primes in linear systems; instead one keeps a single prime-square representation and supplies the remaining multiplicity from the classical sum-of-two-squares function r0, which is accessible by primes in arithmetic progressions on average. On the upper side the same Gaussian factorisation and Selberg sieve as earlier work are retained, but a friable Selberg–Delange estimate is inserted in the medium-prime range so that a Dickman decay factor removes the extra log-log-log that previously spoiled the fourth moment. The same mixed-moment idea also yields a short unconditional proof of the lower bounds for moments of the shifted-prime divisor function.","feed_headline":"Prime-square moments match the predicted size for every k≥4","feed_subtitle":"Unconditional lower bounds replace a Green–Tao conjecture; the fourth-moment logloglog factor is removed","key_machinery":"The mixed moment Mk(x)=∑ R2(n)r0(n)^{k-1}. Hölder together with the known moments of r0 converts a lower bound for Mk into a lower bound for the pure moments of R2; the mixed count reduces to primes satisfying a single congruence m|p^{2}+q^{2}, which is handled by Bombieri–Vinogradov.","core_discovery":"For every fixed integer k≥4 the sum of r2(n)^k over n≤x is asymptotic in order of magnitude to x(log x)^{2^{k-1}-2k-1}. The upper bound for k=4 loses the previous log-log-log factor, the third-moment error reaches the conjecturally optimal O(x/(log x)^3), and all lower bounds hold unconditionally.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["kth moments of two prime-square sums hit predicted order for all k≥4","Unconditional lower bounds settle prime-square moments for every k≥4","Fourth-moment prime-square counts lose the earlier logloglog factor","r2(n)^k sums match true size for fixed k≥4 without Green-Tao conjecture","Correct-order moments for representations by two prime squares, k≥4"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The medium-range upper bound needs a positive-power divisor saving on the singular series over friable bases; if that saving vanished, the Dickman decay alone would not cancel the remaining harmonic sum for k=4.","fun_headline_variants_meta":{"raw":{"variants":["kth moments of two prime-square sums hit predicted order for all k≥4","Unconditional lower bounds settle prime-square moments for every k≥4","Fourth-moment prime-square counts lose the earlier logloglog factor","r2(n)^k sums match true size for fixed k≥4 without Green-Tao conjecture","Correct-order moments for representations by two prime squares, k≥4"]},"model":"grok-4.5","effort":"low","cost_usd":0.005708,"raw_usage":{"total_tokens":1480,"prompt_tokens":697,"num_sources_used":0,"completion_tokens":107,"cost_in_usd_ticks":57080000,"prompt_tokens_details":{"text_tokens":697,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":676,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":697,"tokens_out":107,"duration_ms":7925,"temperature":1.0,"reasoning_tokens":676,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-13T01:13:05.298621+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"If an explicit computation of the fourth moment up to a large x (say 10^{12}) produced a growth rate visibly larger than x/(log x)^5 by more than a slowly growing log-log-log factor, the claimed upper bound would be false.","supporting_citations":[],"review_version":1}