{"id":"03792cb5-eace-4077-bff0-1c902749d881","arxiv_id":"2607.16695","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For A(s)=B(s)=ζ(s)^m with m≥4, the sharply truncated dual sum can be made as large as a positive power of N, disproving Friedlander–Iwaniec Conjecture 1 as stated.","lead":"The paper gives explicit counterexamples to Friedlander–Iwaniec's 2005 conjectured subpower bound for sharply truncated nonlinear dual sums, using the zeta-power datum ζ(s)^m with m≥4. The key idea is simple: choose x so that the last term in the sum sits at a maximum of the cosine, making a whole block of terms add coherently with power-of-N size.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Internal math is sound; the load-bearing risk is external: whether [6] Conjecture 1 includes ζ(s)^m with pole order m≥4 and permits N>x. The paper cites [6] without quoting it; if either fails, the m≥4 headline reduces to the m≥6, N≤x case.","rationale":"The reader's ACCEPT rests on the internal proof being sound and treats the scope as a caveat. I concur that the internal mathematics is correct: Theorem 2.1 is elementary and fully proved, the specialization to ζ(s)^m via the classical functional equation is consistent, and the lower bound contradicts any subpower estimate. However, the headline claim — that Friedlander–Iwaniec Conjecture 1 fails for every m≥4 — depends on two external facts that are asserted but not established by quotation: (1) ζ(s)^m with a pole of order m is within the class of series to which Conjecture 1 applies, and (2) Conjecture 1 carries no N≤x restriction. The paper's own Corollary 1.3(ii) shows the authors are aware that the N≤x range is a possible restriction, and they only handle m≥6 there. Since the entire result is a counterexample to a published conjecture, these scope questions are load-bearing rather than cosmetic. A quick check of the original text will settle them. Until then, the appropriate verdict is conditional acceptance of the construction with the scope caveat, rather than unconditional ACCEPT.","tokens_in":11955,"tokens_out":9704,"duration_ms":97623,"concrete_test":"Retrieve [6] and verify verbatim: (a) the sentences on pp.494–495 that include products of zeta and Dirichlet L-functions and permit poles of arbitrary finite order at s=1; (b) the exact statement of Conjecture 1, checking whether it carries a 1≤N≤x condition or any restriction to cuspidal/simple-pole series; (c) whether κ_j=0 is allowed in [6,(1.4)–(1.8)]. If all three match the paper's usage, the counterexample stands; if any fail, re-evaluate which range of m and N survives.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Theorem 1.2 and the contradiction argument in Corollary 1.3 are correct conditional on the definition of B_{ℓ,D} and on the class of series covered by Conjecture 1.1. The single unverified premise is the exact scope of Friedlander–Iwaniec's Conjecture 1 in [6]. The paper's Conjecture 1.1 (Section 1.3) is a paraphrase, not a quotation; Proposition 3.1 asserts, on the basis of [6, pp.494–495] and [6,(1.4)], that (i) products of zeta with arbitrary multiplicity m are admissible despite the pole of order m at s=1, and (ii) the boundary case κ_j=0 is permitted. Also, the uniform bound (1.9) is stated without the condition 1≤N≤x; if the original conjecture is restricted to N≤x, the m=4,5 counterexamples (with x_N≍N^a, a<1) fall outside that range, leaving only the m≥6, N≤x case of Corollary 1.3(ii). This is not an internal mathematical flaw — I could not find an error in Theorem 2.1 or its specialization — but it is the load-bearing hinge of the headline claim. The paper itself flags related limitations (Remark 2.3, Section 1.5), yet the scope of Conjecture 1 is not quoted and therefore remains an external correctness risk.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to disprove the Friedlander–Iwaniec sharp dual-sum Conjecture 1 by constructing explicit power-sized values of the sharply truncated dual sum for A(s)=B(s)=ζ(s)^m, m≥4. The core is an elementary endpoint-sector lemma (Theorem 2.1): for any phase κ(nx)^ν with 0<ν<1 and coefficients bounded below, one can choose x_N ≍ N^a so that the terminal H_N ≍ N^{1−ν(a+1)} terms lie in a fixed arc where the periodic weight has positive real part, forcing a block contribution ≫ N^{1−β−ν(a+1)}. Specialized with ν=1/m, β=(m+1)/(2m), κ=2πm and c_n=d_m(n), this yields Theorem 1.2 and Corollary 1.3: for m≥4 and 0<a<(m−3)/2, the dual sum is ≫ N^{(m−3−2a)/(2m)} along a sequence, contradicting the conjectured uniform subpower bound. The paper also gives an N≤x counterexample for m≥6 and an N ≍ x^{(m−1)/2} counterexample for m≥5.","tokens_in":12268,"tokens_out":10118,"duration_ms":91602,"significance":"If the Friedlander–Iwaniec conjecture is correctly interpreted to include ζ(s)^m with pole order m≥4 and arbitrary N, the paper's result is a clean, explicit refutation of a published conjecture. The proof is transparent and elementary, with no fitted constants, machine-verifiable computations, and explicit sequences; it also explains why the trivial bound in (1.10) is the best possible in this case. The paper's self-imposed limitations (Remark 2.3, Section 1.5) show awareness of the boundaries of the argument. The main risk is external: the scope of the original conjecture is not quoted, and the counterexample relies on a specific reading of [6]. The m≥6, N≤x part of Corollary 1.3(ii) is robust to the N≤x restriction.","major_comments":[{"comment":"The conjecture is presented as a paraphrase without quoting the original. The paper's claims that [6, pp. 494–495] explicitly allow products of zeta with poles of arbitrary finite order, and that (1.9) is uniform in N without the restriction 1≤N≤x, are load-bearing. If the original conjecture restricts N≤x, the m=4,5 counterexamples (with a<1 so N ≫ x_N) are outside the conjecture's range, leaving only m≥6 with N≤x. If the class of \"automorphic series\" is narrower than the paper's reading (e.g., cuspidal factors with simple poles), ζ(s)^m may not be a permissible datum. Please quote the exact statement of [6, Conjecture 1] and the cited passage, and amend the abstract/theorems if any restriction applies.","section":"Section 1.3, Conjecture 1.1"},{"comment":"The verification that ζ(s)^m lies in the Friedlander–Iwaniec framework relies on the assertion \"Re κ_j=0 is explicitly permitted in [6, (1.4)]\" and \"pole at s=1 of arbitrary finite order [6, pp. 494–495].\" These are not demonstrated in the manuscript. Because the counterexample's validity depends entirely on this inclusion, the authors should reproduce the relevant conditions verbatim and confirm that the boundary case κ_j=0 and pole order m are allowed. If they are not, the counterexample does not lie in the class covered by the conjecture.","section":"Section 3.1, Proposition 3.1"}],"minor_comments":[{"comment":"Typo: \"Organiztion\" should be \"Organization.\"","section":"Section 1.6"},{"comment":"Grammar: \"Let a(n) and b(n) are arithmetic sequences\" should be \"Let a(n) and b(n) be arithmetic sequences.\"","section":"Abstract"},{"comment":"The condition \"1≤N≤x\" appears in (1.8) but is omitted in Conjecture 1.1; clarify whether the conjecture is stated for all N or for the same range, and ensure the notation B_{\\ell,D}(x,N) is consistent.","section":"Section 1.3"},{"comment":"The statement \"This is only an amplitude-level statement...\" is useful, but it might be placed earlier to avoid confusion about the role of Clozel–Sarnak's framework.","section":"Remark 2.3"}],"recommendation":"major_revision","confidential_remarks":"The mathematical core is sound and the counterexample appears genuinely striking. The revision is needed to nail down the scope of the conjecture being refuted; if the author can supply exact quotes showing inclusion, this becomes a strong accept. The paper explicitly acknowledges the N≤x issue in Corollary 1.3(ii), so the revision is straightforward. I suggested major_revision rather than minor because the abstract's claim m≥4 may need to be qualified, which is a substantive correction."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this paper for one thing: it gives a very simple, explicit construction that appears to kill Friedlander–Iwaniec's Conjecture 1 for A(s)=B(s)=ζ(s)^m, m≥4. The mechanism is just resonance at the endpoint of the truncated sum: choose x so the phase of the last term sits at a cosine maximum, then the block of about N^{1-(a+1)/m} consecutive terms all land in a fixed positive arc, and since d_m(n)≥1 the weighted mass is N^{(m-3-2a)/(2m)}. That is the whole argument, and it works. I checked Theorem 2.1 carefully and it is correct; the specialization to the zeta power is consistent with the classical functional equation and the coefficient bound on d_m(n) is fine. If the conjecture is open to zeta-powers with boundary spectral parameter zero, this is a genuine counterexample, and the paper is honest about not touching Conjecture 2.\n\nThe real soft spot is external, not internal. The paper paraphrases the Friedlander–Iwaniec conjecture rather than quoting it. Conjecture 1.1 in the paper asserts the bound for 'automorphic series' in their framework, and Proposition 3.1 claims ζ(s)^m qualifies citing [6, pp. 494–495]. But if the original statement is restricted to cuspidal L-functions with simple poles, or to N≤x, then the m=4,5 counterexamples with x_N≍N^a, a<1 fall outside that range, and the headline reduces to the m≥6, N≤x case of Corollary 1.3(ii). That is still a counterexample, but a thinner one. This is a scope caveat, not a flaw in the math; the paper itself flags related limitations.\n\nI also found the paper's comparison with prior resonance and Ω-result literature fair, and the self-contained proof is refreshingly short. The internal reasoning is sound enough that I'd bet on the counterexample surviving, modulo the scope question.\n\nWho is this for? Anyone working on nonlinear dual sums, Voronoi summation, or the Piltz divisor conjecture. It deserves a serious referee: a referee should check the original Friedlander–Iwaniec text and decide whether the conjecture indeed covers zeta-powers. If it does, this is a real result. I'd send it out.","headline":"A clean, elementary endpoint-resonance construction that likely refutes Friedlander–Iwaniec's uniform subpower bound for dual sums of ζ(s)^m, with the main uncertainty being the exact scope of the original conjecture.","tokens_in":12775,"tokens_out":1701,"would_cite":true,"duration_ms":18108,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M41","11L07","11N37","11F66"],"pacs":[],"model":"deepseek-v4-flash","headline":"The Friedlander–Iwaniec dual-sum conjecture fails for every zeta power of degree m≥4.","keywords":["Friedlander–Iwaniec conjecture","nonlinear dual sums","sharp truncation","zeta function powers","divisor function","endpoint resonance","subpower bounds","exponential sums"],"falsifier":"Consult the original statement of the conjecture to verify that a degree-m Dirichlet series with a pole of order m≥4 at s=1 and κ_j=0 is explicitly admitted; alternatively, numerically compute B_4(x_N,N) for m=4, a=0.4, λ=1 for large N and check whether its magnitude consistently exceeds, say, N^{0.02} as predicted.","tokens_in":11798,"feed_emoji":"⚡","tokens_out":6098,"duration_ms":55617,"temperature":0.7,"pith_summary":"This paper constructs explicit counterexamples to a 2005 conjecture predicting that sharply truncated nonlinear dual sums of L-function coefficients are bounded by a subpower factor uniformly in the truncation point and the external variable. Taking both Dirichlet series to be the m-th power of the Riemann zeta function with m≥4, the author tunes the external variable x so that a block of the last H terms near the cutoff N all have the same sign. Because the m-fold divisor numbers satisfy d_m(n)≥1, this block has mass N^Δ with Δ>0, contradicting the conjectured N^ε bound. The failure persists even in the printed range N≤x for m≥6.","feed_headline":"Zeta-power sums refute a dual-sum conjecture","feed_subtitle":"For every m≥4, resonant truncations of ζ(s)^m grow like N^positive power, not N^ε.","key_machinery":"The sharp dual sum B_{ℓ,D}(x,N) and a general endpoint-sector lemma (Theorem 2.1). For a phase Φ(n)=κ(nx)^ν with 0<ν<1, the derivative at n=N is of size N^{ν(a+1)-1} when x≍N^a, so the phase changes by only a bounded amount on a block of reciprocal-derivative length H≍N^{1-ν(a+1)}. Choosing x so the terminal phase hits a non-zero point of the periodic profile forces the whole block to align; with coefficients bounded below by c*>0, the block's weighted sum is power-sized. Applied to ζ(s)^m, the coefficients are d_m(n)≥1, ν=1/m, and the profile is the cosine.","core_discovery":"Theorem 1.2 and Corollary 1.3: for every integer m≥4 there exist x_N≍N^a with 0<a<(m-3)/2 and M_N∈{N-H_N,N} such that |B_m(x_N,M_N)| ≫ N^{(m-3-2a)/(2m)}. Since the exponent is positive, the uniform subpower bound B_{ℓ,D}(x,N)≪(DNx)^ε in the Friedlander–Iwaniec conjecture fails for A(s)=B(s)=ζ(s)^m. The mechanism is endpoint resonance: choosing x so that the phase 2πm(nx)^{1/m}+φ_m is at a cosine maximum when n=N makes the phase vary by O(1) over a block of length H ≍ N^{1-(a+1)/m} preceding N, and all those terms contribute with the same sign.","pith_inferences":["The endpoint-sector lemma suggests that any L-function with a pole of order at least 4 at s=1 (or otherwise yielding coefficients bounded below) can be used to build counterexamples; one might test this for products of Dirichlet L-functions with multiple poles.","The conjecture may remain plausible for cuspidal automorphic forms whose coefficients are not bounded below by a positive constant (e.g., Hecke eigenvalues with sign changes); the lower bound d_m(n)≥1 is essential to the construction.","A quick numerical check for m=4, a=0.4, N=10^6 using the explicit formula (3.4) should show |B_4| growing like N^{1/40}, which would settle the matter concretely."],"forward_implications":["The uniform subpower estimate in Friedlander–Iwaniec Conjecture 1 is false for the zeta-power datum; any proof would have to exclude such poles or restrict the class of series.","The counterexample satisfies N≤x for m≥6, so relaxing or enforcing that condition does not save the conjecture in the zeta-product case.","The proposed deduction of the omega-free error-term conjecture (Conjecture 2) from Conjecture 1 via N≍x^{(m-1)/2} collapses for these examples, though Conjecture 2 itself is not disproved.","Any sharp-cutoff sum with positive coefficients bounded below by a constant and a slowly varying nonlinear phase will exhibit the same resonance; the obstruction is not peculiar to zeta powers."],"fun_headline_variants":["Zeta-power sums break dual-sum conjecture","Resonance in zeta powers defeats subpolynomial bound","Zeta powers disprove dual-sum subpolynomial bound","Dual sums of zeta powers grow, counter to conjecture"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The counterexample applies only if ζ(s)^m, with its pole of order m at s=1 and boundary spectral parameters κ_j=0, is an admissible automorphic series in the original conjecture; the paper asserts this on the basis of the original framework, but if that framework secretly requires cuspidal L-functions with simple poles, the counterexample would lie outside its scope.","fun_headline_variants_meta":{"raw":{"variants":["Zeta-power sums break dual-sum conjecture","Resonance in zeta powers defeats subpolynomial bound","Zeta powers disprove dual-sum subpolynomial bound","Dual sums of zeta powers grow, counter to conjecture"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000887,"raw_usage":{"total_tokens":3746,"prompt_tokens":907,"completion_tokens":2839,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":651,"completion_tokens_details":{"reasoning_tokens":2773}},"tokens_in":651,"tokens_out":2839,"duration_ms":21911,"temperature":1.0,"reasoning_tokens":2773,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T20:14:00.664397+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Consult the original statement of the conjecture to verify that a degree-m Dirichlet series with a pole of order m≥4 at s=1 and κ_j=0 is explicitly admitted; alternatively, numerically compute B_4(x_N,N) for m=4, a=0.4, λ=1 for large N and check whether its magnitude consistently exceeds, say, N^{0.02} as predicted.","supporting_citations":[],"review_version":1}