{"id":"62925e13-322d-4f0f-8563-367c3d49bf4c","arxiv_id":"2603.24120","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Assuming GRH, the average number of representations of multiples of q as sums of two prime k-th powers is asymptotically a constant depending on q and k times the unrestricted average, with a stated error term.","lead":"This paper gives a formula for the average number of ways a multiple of q can be written as a sum of two prime k-th powers, assuming the Generalized Riemann Hypothesis. It generalizes an earlier result for the prime case k=1 and finds that the main term sometimes vanishes, depending on q and k.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The I2 estimate in §3.3 rests on an unproved adaptation of Gallagher's lemma to n^k-phase sums; the proof as written does not establish the O(N^{1/k} log^2 N) error of Theorem 1.3.","rationale":"The paper's main-term setup is plausible: the character sum Σ_k(q) and the decomposition into I1, I2, I3 are natural, and the evaluation of Σ_k(q) is essentially correct apart from a fixable parity slip in the q=4, k odd case. The make-or-break point is the estimate of I2 in §3.3. The application of Gallagher's lemma to the sum with phase n^k is the single place where a new, unproved ingredient is introduced. Every later estimate (J1, J2, the dyadic sum) is expressed in the variable n and presupposes that short intervals over n appear. This is not a matter of tightening constants: the standard lemma has short intervals x<n≤x+h only because the phase is nx, and the counterexample with alternating coefficients and h=2 shows that the same inequality with n^k phases and n-intervals is false. Even if one repaired it by applying Montgomery in the variable n^k, the paper's own J1/J2 lemmas would need to be reformulated, and the claimed error term would not follow from the current text. The dropped h^2 log^2(2q) term is a second, independent sign that the I2 calculation is internally inconsistent. I therefore agree with the reader's rejection: the central asymptotic may be true, but this manuscript does not prove it.","tokens_in":12376,"tokens_out":24371,"duration_ms":231909,"concrete_test":"Verify the Gallagher-type inequality used in §3.3. A direct test: take k=2, h=2, a_n=(-1)^n for 1≤n≤M with M even. The RHS 1/h^2 ∫ |∑_{x<n≤x+h} a_n|^2 dx is O(1) (only boundary intervals contribute), while ∫_0^{1/(2h)} |∑ a_n e(n^2 α)|^2 dα grows linearly in M (numerically ≈ M/4 for large M). If confirmed, the adapted lemma as stated is false and the I2 estimate must be redone. Alternatively, re-derive the step from Montgomery's Lemma 1.9: replacing e(nx) by e(n^k x) requires summing over x < n^k ≤ x+h, not over x < n ≤ x+h.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.3 applies Montgomery's Gallagher lemma (Lemma 1.9) to Ψ_k(z,χk)=∑ χk(n)Λ(n)e^{-n^k/N}e(n^k α), whose frequencies are n^k, but writes the short interval as ∑_{x<n≤x+h} χk(n)Λ(n)e^{-n^k/N}. The standard lemma is for ∑ a_n e(nx) with intervals in the summation index n; for frequencies n^k the proof does not go through because the phase difference is n^k−m^k, not n−m. This is load-bearing: the entire bound for I_m, hence for I2 in (3.2)–(3.13), depends on this step. If one instead applies Gallagher in the variable n^k, the subsequent J1/J2 estimates in n do not apply. No proof or reference is supplied. Independently, even granting this step, (3.13) contains a term h^2 log^2(2q); after division by h^2 this contributes log^2(2q) per dyadic block, which is not dominated by N^{1/k}/h log^2 N when h is large (e.g., k=2, h≫N^{1/2}), and summing over m yields an unacceptable N log^2 q term. Thus the proof as written does not yield the claimed error term.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims Theorem 1.3: for fixed q≥2 and k≥2, assuming GRH for modulus q, the average G_{q,k}(N) of the number of representations of n≤N, q|n, as a sum of two prime k-th powers equals (Σ_k(q)/φ(q)) G_{1,k}(N) plus an error term O(N^{1/k} log^2 N log q / φ(q)). The proof uses a Dirichlet character decomposition of the generating function, an explicit formula for Ψ_k(z,χ) under GRH, estimates for character sums in short intervals, and an explicit evaluation of the constant Σ_k(q). This generalizes the k=1 result of Ikeda and Suriajaya.","tokens_in":12738,"tokens_out":19257,"duration_ms":155851,"significance":"If the result holds, it is a natural and worthwhile generalization of the Goldbach-type average over multiples of q to prime powers, and it uncovers a new phenomenon: the proportionality constant Σ_k(q) can vanish, so the restricted average may be of lower order than the unrestricted one. The explicit evaluation of Σ_k(q) in Lemma 4.1 is a concrete, self-contained contribution. However, the proof as written contains several load-bearing gaps, especially in the short-interval estimates of Section 3.3–3.5. These make the current version unsuitable for publication until the technical issues are resolved.","major_comments":[{"comment":"The proof applies Montgomery's Gallagher lemma (Lemma 1.9 of [7]) to Ψ_k(z,χ)=∑ χ^k(n)Λ(n)e^{-n^k/N}e(n^kα), whose frequencies are n^k. The standard lemma applies to ∑ a_n e(nx) with short intervals in the index n. The paper writes the short interval as ∑_{x<n≤x+h} χ^k(n)Λ(n)e^{-n^k/N}, which is not what the standard lemma yields. A change of variable to m=n^k would give intervals in m, not in n. No statement or proof of the needed adaptation is provided. This is load-bearing for the entire I2 estimate in (3.2)–(3.13). Please supply a proof of the claimed variant, or replace this step with a valid large-sieve bound for polynomial phases.","section":"§3.3"},{"comment":"Even granting the Gallagher lemma adaptation, the estimate (3.13) gives I2(N,h)≪hN^{1/k}log^2N + h^2log^2(2q). After dividing by h^2 and substituting into the dyadic sum in (3.2), the h^2log^2(2q) term contributes N log^2 q (since the outer weight is N/2^m and summing over m gives N log^2 q). This is not dominated by N^{1/k}log^3N for k≥2. The sentence 'q-terms do not appear because q is fixed and thus its contribution can be neglected' is not valid: a constant factor is not negligible when it is not multiplied by a term that vanishes. Thus the claimed bound I2=O(N^{1/k}log^3N) is not established.","section":"§3.5, Eq. (3.13)"},{"comment":"The main term I1 is defined in (2.7) with |Ψ_k(z,χ0)|^2, but Lemma 2.1 gives a product Ψ_k(z,χ^k)Ψ_k(z,\\bar{χ}^k), which for χ^k=χ0 equals Ψ_k(z,χ0)^2, not its absolute square. In (3.1), the paper replaces this by (Ψ_k(z)+O(...))^2, which is only consistent with the non-absolute square. If the absolute value is retained, the integral does not equal G_{1,k}(N) because the phase in the generating function is additive (n^k+m^k), not the difference. Please correct the notation and the main-term derivation.","section":"§3.2, Eqs. (2.7) and (3.1)"},{"comment":"The proof concludes at the end of §3.5 that I2=O(N^{1/k}log^3N). This does not match the error term in Theorem 1.3, which is O(N^{1/k}log^2N log q/φ(q)). For fixed q, log^3N is not O(log^2N). Even if the rest of the proof were correct, the final error would be at least O(N^{1/k}log^3N). Either the theorem's error term must be weakened to O(N^{1/k}log^3N), or the I2 estimate must be improved by a factor log N.","section":"Theorem 1.3 vs. §3.5"},{"comment":"The proof of Lemma 3.1 for the case where χ^k is not primitive is deferred to 'arguing like in the proof of Lemma 2.4 of [4]' without providing the details. This lemma is essential for the estimates of J1 and J2, which directly feed into the short-interval bounds. Since the case χ^k=χ0 is excluded, but χ^k may still be non-primitive, a complete proof is needed. Please supply the full argument or a precise reference with the necessary modifications.","section":"Lemma 3.1"}],"minor_comments":[{"comment":"Reference [1]: 'Springler-Verlag' should be 'Springer-Verlag'.","section":"References"},{"comment":"In the statement for p=2, α≥3, the condition '3≤β<α−2' appears to be a typo. For example, β=1 or 2 with α≥4 should also be covered; the proof text uses '1≤β<α−2'. Please correct the range.","section":"Lemma 4.1, p=2 case"},{"comment":"In the integration by parts after (3.11), the function J1(x) is defined using ψ_k(x,χ), while the integral contains ψ_k(x+h,χ). The integration by parts should involve the shifted version J1^{(h)}(x)=∫_0^x |ψ_k(u+h,χ)|^2 du, not J1(x). This is not justified and needs correction.","section":"§3.5, Eq. (3.11)"},{"comment":"The displayed bound 'I2 ≪ N^{1/k} log^3 N log^2 q < N^{1/k+1}' is confusing; the first inequality is far weaker than the final claim in Theorem 1.3. Please align the sketch with the actual proof.","section":"Sketch of proof"}],"recommendation":"major_revision","confidential_remarks":"I agree with the reader's report that the proof has serious gaps, particularly the unproved adaptation of Gallagher's lemma. This is not a small technicality: the short-interval estimates in Section 3.3 are the core of the I2 bound. The mismatch of the error term between the theorem and the proof is also concerning. Nevertheless, the main idea is plausible and the computation of Σ_k(q) is a useful result. I would not reject outright because the central theorem may be salvageable with substantial revision. The authors should be asked to (a) prove a correct short-interval estimate for n^k-phase sums, (b) fix the log-power mismatch in the final error, (c) clarify the |Ψ|^2 vs Ψ^2 issue, and (d) provide full details for Lemma 3.1."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Take a look at this one. The paper extends Ikeda and Suriajaya's average Goldbach theorem from k=1 to prime k-th powers, and the main term involves a character sum Σ_k(q) over χ with χ^k=χ_0. That is genuinely new, and the evaluation of Σ_k(q) in Lemma 4.1 looks correct and is a nice use of the cyclic group structure. If the main theorem is true, the constant can vanish, which is an interesting phenomenon worth understanding.\n\nThe proof of the error term, though, doesn't hold up. The I2 estimate in Section 3.3 has two problems. First, Gallagher's lemma is applied to a sum with phase n^k α, with short intervals in n, but the standard lemma is for e(nα) and intervals in the summation variable. Going through the n^k variable would give intervals in n^k, and the J1/J2 estimates in n no longer apply. The paper supplies no proof of this adaptation. Second, even granting that, the bound (3.13) contains an h^2 log^2(2q) term. Dividing by h^2 and summing over dyadic blocks leaves a log^2 q term per block, and the sum over m gives N log^2 q. For k≥2 that is far larger than the claimed N^{1/k} log^2 N. The proof just drops it. So Theorem 1.3 as stated is not established.\n\nThe rest of the paper is more solid. Lemma 3.1 (the J1/J2 bounds for χ^k) is a standard adaptation, and the I1 evaluation is fine. The q=4 calculation is fine. The issue is localized to Section 3.5, which is good news because it might be repairable—a more careful application of Gallagher in the n^k variable and a sharper treatment of those extra terms could yield a weaker but still acceptable error. But the current version's central theorem is unsupported.\n\nThis is a paper with a good idea and a real gap. I'd send it to a referee, because the result is worth getting right and the Σ_k(q) evaluation is valuable on its own. But the current version should not be accepted as is.","headline":"Genuinely new generalization with a nice character sum, but the I2 error estimate has two concrete gaps that leave Theorem 1.3 unproven as written.","tokens_in":13202,"tokens_out":5704,"would_cite":false,"duration_ms":51113,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11P32","11M26"],"pacs":[],"model":"deepseek-v4-flash","headline":"For fixed q and k, the average number of representations of an integer as a sum of two prime k-th powers, taken over multiples of q, is a constant multiple of the unrestricted average — and that constant can be zero.","keywords":["prime powers","average number of representations","arithmetic progressions","Dirichlet characters","Generalized Riemann Hypothesis","character sums","sums of two prime powers"],"falsifier":"Compute I_2(N,h) directly for a small case, say k = 2, q = 3, and a non-principal character χ with χ^2 ≠ χ_0, using the standard Gallagher lemma on the sequence n^2 instead of n, and compare the resulting bound with the claim I_2 ≪ N^{1/k} log^3 N. A divergence would indicate that the short-interval adaptation in Section 3.3 is invalid.","tokens_in":12228,"feed_emoji":"🔢","tokens_out":2311,"duration_ms":23363,"temperature":0.7,"pith_summary":"This paper studies the average, over integers n ≤ N divisible by a fixed q, of the number of ways n can be written as a sum of two prime k-th powers. It proves that, assuming the Generalized Riemann Hypothesis, this average is asymptotically a constant multiple of the same average over all integers. The new constant, Σ_k(q), is an explicit character sum that depends on k and q, and in some cases it vanishes, meaning the expected main term disappears entirely. This extends a known result for sums of two primes (k=1) to all k ≥ 2, and it shows that k-th powers are not uniformly distributed in arithmetic progressions in this average sense.","feed_headline":"Prime-power sums over multiples of q obey a constant ratio","feed_subtitle":"The new constant Σ_k(q) can vanish, making the main term disappear for some moduli.","key_machinery":"The proof decomposes the generating function F_{q,k}(z) into Dirichlet characters, reducing the problem to exponential sums Ψ_k(z, χ) = Σ_n χ(n)Λ(n) z^{n^k}. An explicit formula, assuming GRH, expresses Ψ_k(z, χ) in terms of a sum over non-trivial zeros of L(s, χ). The main term is isolated as the contribution of characters satisfying χ^k = χ_0, producing the character sum Σ_k(q). The error term is controlled using Gallagher's lemma applied to short intervals of n, together with estimates for the mean-square of the summatory function ψ_k(x, χ).","core_discovery":"Theorem 1.3: For fixed q ≥ 2 and k ≥ 2, under GRH for Dirichlet L-functions modulo q, the average G_{q,k}(N) satisfies G_{q,k}(N) = Σ_k(q)/φ(q) · G_{1,k}(N) + O(N^{1/k} log^2 N log q / φ(q)), where Σ_k(q) = Σ_{χ^k = χ_0} χ(−1). The main term is therefore a constant multiple of the unrestricted average G_{1,k}(N), and the constant is an explicit character sum that can vanish. When Σ_k(q) = 0, the average over multiples of q is of smaller order than the unrestricted average, illustrating that k-th powers are not uniformly distributed among residue classes modulo q.","pith_inferences":["A testable extension is to consider j ≥ 3 prime k-th powers; the paper mentions such work in progress, but the same character-sum mechanism would likely produce a constant multiple involving a higher-order character sum.","The vanishing of Σ_k(q) suggests that the equation m_1^k + m_2^k ≡ 0 mod q with (m_1 m_2, q) = 1 has no solutions when Σ_k(q) = 0; counting such solutions directly for small q and k would provide a concrete check of Lemma 4.1.","One could ask whether the error term O(N^{1/k} log^2 N log q / φ(q)) is sharp; if a sharper lower bound for I_2 could be proved without the adapted Gallagher step, it would confirm or refute the validity of that adaptation.","The proportionality constant being possibly zero may have consequences for sieve or circle-method treatments of prime powers in arithmetic progressions: the expected main term can be rescued only by characters with χ^k = χ_0."],"forward_implications":["If the theorem is correct, the average number of representations over multiples of q is exactly proportional to the unrestricted average, with proportionality factor Σ_k(q)/φ(q).","For cases where Σ_k(q) = 0 (e.g., q = 3, k = 2), the main term vanishes, so the average over multiples of q is asymptotically smaller than the unrestricted average.","The result extends the k = 1 theorem of Ikeda and Suriajaya to all k ≥ 2, with a refined error term that depends on q only through log q / φ(q).","The evaluation of Σ_k(q) gives a complete multiplicative formula, showing exactly when the constant is zero, one, or φ(p^α)-related, depending on the greatest common divisor of k and φ(p^α)."],"fun_headline_variants":["Prime k-th power sums over multiples of q: constant can vanish","When Σ_k(q)=0, main term vanishes in prime-power sum averages","Multiples of q distort prime k-th power representation averages","Vanishing constant changes order for sums of two prime powers"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof assumes that Gallagher's lemma, applied in Section 3.3 to the exponential sum whose phase is n^k α, may be written with short intervals over n (x < n ≤ x+h) rather than over the frequencies n^k; the paper gives no proof of this adaptation, and if it fails the bound for I_2 collapses.","fun_headline_variants_meta":{"raw":{"variants":["Prime k-th power sums over multiples of q: constant can vanish","When Σ_k(q)=0, main term vanishes in prime-power sum averages","Multiples of q distort prime k-th power representation averages","Vanishing constant changes order for sums of two prime powers"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000114,"raw_usage":{"total_tokens":831,"prompt_tokens":597,"completion_tokens":234,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":341,"completion_tokens_details":{"reasoning_tokens":173}},"tokens_in":341,"tokens_out":234,"duration_ms":3143,"temperature":1.0,"reasoning_tokens":173,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T17:35:18.210237+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute I_2(N,h) directly for a small case, say k = 2, q = 3, and a non-principal character χ with χ^2 ≠ χ_0, using the standard Gallagher lemma on the sequence n^2 instead of n, and compare the resulting bound with the claim I_2 ≪ N^{1/k} log^3 N. A divergence would indicate that the short-interval adaptation in Section 3.3 is invalid.","supporting_citations":[],"review_version":1}