{"id":"3c2ad686-02d8-476b-a0cc-7de48acd4fc3","arxiv_id":"2510.27444","paper_version":4,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":5,"one_line_summary":"An improved explicit zero-counting bound for Dedekind zeta functions is proposed, but the final constant contradicts the proof's own derivation.","lead":"This paper claims a new method for counting non-trivial zeros of Dedekind zeta functions up to height T, with explicit error terms that would improve earlier bounds for number fields and for the Riemann zeta function. The main theorem's stated constant does not match the algebra in the proof, so the claim as written is not established.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1 hinges on Lemma 2.1's H≤0, which is not rigorously proved: the boundary check reduces to a Mathematica claim about the seven real roots of a degree-13 polynomial, with no interval certificate.","rationale":"After working through the proof, the most load-bearing assumption is the numerical certifier in Lemma 2.1. The reader's weakest_assumption points to this, and I agree. The final concatenation step that the reader flags as a false equality is actually correct: expanding log(d_K(T/(2π))^{n_K}) turns the coefficient 5.899 into 5.543 because 0.194 log(2π)≈0.356. So that objection should be retracted. But the central theorem still cannot be accepted because the H≤0 inequality is not proven to the standard required for an explicit bound. The proof's own constants at the local maxima are around -2×10⁻⁴, so this is not a qualitative remark; a tiny error in the root computation could flip the sign. The same style of black-box numerics recurs in Lemma 2.7, where '(2.14) holds for all α≥4' is asserted after a Mathematica computation and the tail bound '≤4.5243' is not backed by visible calculations. Since the error terms improve on previous work by more than a factor of four in the n_K coefficient, the numerical constants are the entire content. Until these are certified by interval arithmetic or a machine-checked proof, the correct verdict is unchanged: reject (or at best conditionally accept) the paper in its current form. My recommendation is UNCHANGED because the reader's reject verdict remains appropriate; the essential defect is the same one the reader identified, even though one of the reader's supporting arguments was mistaken.","tokens_in":16645,"tokens_out":33769,"duration_ms":260294,"concrete_test":"Run a fully rigorous verification of Lemma 2.1: define h(t)=H(1/2,t) with d=0.722, a1=1.07, a2=0.93, a3=0.365. Compute h'(t), clear denominators to obtain a degree-13 polynomial P(t). Use Sturm's theorem or a validated root-isolation package (e.g., Mathematica's RootInterval, arb, or Sage with RealIntervalField) to certify that P has exactly seven real roots, say in disjoint intervals containing the claimed values. On each interval, evaluate h(t) with interval arithmetic to show h(t)≤0 (or, at critical points, that the local maxima are <0). Also choose an explicit T0, e.g. T0=20, and use uniform interval bounds in b∈[-1/2,1/2] from the asymptotic expansion to verify H(b,T0)≤0. If every interval evaluation is negative, Lemma 2.1 is established; if any interval contains a positive value, the central estimates collapse.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 2.1 bounds f(b,t,d) by an expression involving da1, d²a2, a3. The proof reduces the problem to showing H(b,t)≤0 on the rectangle boundary. For the vertical sides at b=±1/2 it studies h(t)=H(1/2,t). The argument says h'(t) has zeros that are roots of a degree-13 polynomial when multiplied out, that there are seven real roots with the listed locations, and that the local maxima have numerically small negative values (≈−0.00019, −0.00022, −0.00015). No Sturm sequence, interval-arithmetic evaluation, or machine-checked certificate is provided. Because the margin at the local maxima is only about 10⁻⁴, a small rounding error would change the sign. Moreover, the asymptotic argument only invokes an unspecified T0 for which H(b,T0)≤0 is asserted uniformly in b; the existence of such T0 is plausible but never made explicit. If H is positive anywhere, the bounds on ∑f, hence E2/E3, and ultimately the constant 5.543n_K in Theorem 1.1 fail. The same lack of rigorous numerical certification appears in Lemma 2.7 (e.g., 'following a mathematica computation' for (2.14) and the tail bound). Note: the reader's objection to the equality 5.899n_K=5.543n_K is not valid, since 0.194 log(d_K(T/(2π))^{n_K})=0.194(log d_K+n_K logT)−0.356n_K; the numerical gaps, not this expansion, are the real obstacle.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a new explicit estimate for the number N_K(T) of nontrivial zeros (counted with multiplicity, with boundary weights) of a Dedekind zeta function up to height T. The method follows Turing's idea: after an argument-principle identity, the zero sum is bounded through a two-sided harmonic majorant f(b,t,d) depending on parameters d,a1,a2,a3, and the resulting logarithmic-derivative expressions are estimated factor-by-factor for the completed zeta function. The main theorem claims |N_K(T) - (T/pi) log(d_K (T/(2pi e))^{n_K}) - 1.919| ≤ 0.194(log d_K + n_K log T) + 5.543 n_K + 0.462 for T≥1, and Corollary 1.2 states the analogous Riemann-zeta bound with constants 0.097 and 3.962. The proof is conditional on two substantial numerical assertions: Lemma 2.1, which reduces to a claim about the sign of a function H(b,t) supported by approximate roots of a degree-13 polynomial, and Lemma 2.7, which uses several Mathematica range computations and a first-22-prime summation without certified interval arithmetic.","tokens_in":17085,"tokens_out":22011,"duration_ms":169625,"significance":"If all numerical claims could be made fully rigorous, the announced constants would be a genuine improvement over the comparable explicit bound of Hasanalizade–Shen–Wong (0.228, 23.108, 4.52) and would improve the known Riemann-zeta zero-counting estimate in the same style. The structural idea of replacing the zero-sum by a carefully optimized harmonic majorant is attractive and appears to be adaptable to other L-functions. However, the significance is currently conditional: the two main lemmas rest on non-certified computer computations, and the margins in the key inequality are as small as 10^-4, so the result is not yet established to the standard required for an explicit-number-theory paper. The paper does not ship machine-checked proofs or interval-verified certificates; it only points to an arXiv page for a Mathematica notebook.","major_comments":[{"comment":"This lemma is the hinge of the proof: it bounds the zero-summand f(b,t,d) and thereby controls E2 and E3. The proof reduces the boundary check to h(t)=H(1/2,t) and then states that h'(t) has zeros which are the roots of a degree-13 polynomial, that there are seven real roots, and that the local maxima have numerically small negative values h(t2)≈-0.00019, h(t4)≈-0.00022, h(t6)≈-0.00015. No Sturm sequence, interval-arithmetic enclosure, or machine-checked certificate is provided. Because the margins at the local maxima are of order 10^-4, ordinary floating-point evaluation is not a rigorous proof; a rounding error of that size would change the sign and invalidate Lemma 2.1, hence also the bounds on E2,E3 and Theorem 1.1. In addition, the appeal to 'large |t|' via (2.9) leaves T0 unspecified and the O-term unquantified. This is a load-bearing gap and must be repaired by certified computati","section":"§2, Lemma 2.1 (Eqs. (2.8)–(2.9))"},{"comment":"Lemma 2.7 estimates E_u(ζ_K) through q1 and concludes E_u(ζ_K)≤5.633 n_K. The proof relies on non-certified Mathematica assertions: the statement that max_x q1(α,x)≤max_x f2(α,x) for all α≥4 ('Following a mathematica computation...', Eq. (2.14)), the numerical evaluation of the sum over the first 22 primes (≤1.1084), and the tail estimate (4.5243). The same kind of unverified 'computing the range with mathematica' steps appears in Lemma 2.6 (U_1,1+U_B1, L_1,1+L_B1, etc.). These numerical inputs determine the additive constants in Theorem 1.1; without interval-verified bounds or exact inequalities, the theorem is not established. A link to the arXiv abstract page is not a substitute for an included, verified computation. This issue is fixable in principle, but it is load-bearing and must be addressed.","section":"§2, Lemma 2.7 (Eq. (2.14) and Eq. (2.16)) and Lemma 2.6"}],"minor_comments":[{"comment":"The asymptotic in (2.9) appears to omit a factor d^2: a direct expansion gives H(b,t)= -π d^2(a1-a2)/(2t^2)+O(t^{-3}), not π/(2t^2)(a2-a1)+O(t^{-3}). The sign is still negative, but T0 should be chosen with the correct coefficient and the O-term should be quantified.","section":"§2, Lemma 2.1"},{"comment":"The constants in the combining paragraph do not match Lemmas 2.6 and 2.7: 5.633+0.258=5.891, not 5.899, and the lower-side coefficient is 5.633+0.25=5.883, not 5.891. The displayed inequalities should be corrected and the final arithmetic recomputed. I note that the subsequent equality 0.194 log(d_K(T/(2π))^{n_K}) = 0.194(log d_K+n_K log T)-0.356 n_K is arithmetically correct; the problem lies in the inconsistent intermediate constants, not in that rewrite.","section":"§2, final combining paragraph"},{"comment":"The text states 'M=p_1000=79'; this should be M=p_22=79. Also, the reference to 'equation (2)' near the end of the proof is dangling, since no equation (2) is numbered.","section":"§2, Lemma 2.7"},{"comment":"For reproducibility, the Mathematica notebook should be included as a supplement with version information. Better still, the decisive numerical inequalities should be certified with interval arithmetic or exact root isolation.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"I am not recommending rejection because the core strategy appears sound and the missing rigor is, in principle, repairable within the scope of a revision. The main risk is that Lemma 2.1's boundary inequality has very small margins, so any successful revision must provide a genuinely certified computation, not merely a new version of the same floating-point plot. The paper's use of the HSW21 bound is not circular; it is only used to justify convergence of the zero sum. I would encourage the editor to ask for a revised version with machine-verified certificates for Lemmas 2.1, 2.6 and 2.7 before considering acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nRead the Amberger paper on counting zeros of Dedekind zetas. The core idea is new: an operator-based decomposition, inspired by Turing's method for S1(T), that lets the author bound the zero-counting error term through a cleverly chosen 'pole' combination. The claimed constants, e.g., 0.194(log d_K + n_K log T)+5.543 n_K+0.462, and the Riemann corollary 0.097 log T+3.962, are indeed improvements over HSW21 and earlier work. If they hold, this is a solid contribution to explicit analytic number theory.\n\nThe structure is clean: decompose the completed zeta, estimate the polynomial, discriminant, gamma, and zeta pieces separately. The gamma estimates via Binet are standard. The zeta-piece estimate uses an Euler-product bound and a prime-ideal splitting argument. The author also provides a Mathematica notebook, which is more than we usually get.\n\nThe main problem is Lemma 2.1. The whole theorem hinges on showing H(b,t) ≤ 0, and the proof does not do that rigorously. For the vertical sides, the argument reduces to a function h(t) whose derivative's zeros are roots of a degree-13 polynomial; the author states that 'multiplying out' gives seven real roots, and that the local maxima are negative but only by about -0.0002. No Sturm sequence, interval-arithmetic certificate, or explicit T0 is given. That margin is too close to trust floating point. If H is positive anywhere, the bounds on Σf, and hence the main theorem, collapse. Lemma 2.7 has the same issue: 'following a mathematica computation' is used to claim a key inequality (2.14).\n\nOne specific reader objection is wrong: the final line equating 0.194 log(d_K (T/(2π))^{n_K}) + 5.899 n_K with 0.194(log d_K + n_K log T) + 5.543 n_K is correct—the log(2π) term accounts for the difference. So the theorem as stated is not internally contradictory on that page. The problem is the unverified numerical lemma, not that arithmetic.\n\nMinor points: Lemma 2.3 asserts 'no local extrema' without proof; probably true but should be checked. The Riemann corollary could compare to modern S(T) bounds but that omission is not serious.\n\nVerdict: the method deserves a serious referee. Send it to review, but the referee should demand a certified verification of the key lemma—interval arithmetic, Sturm sequences, explicit T0—and a similar treatment of Lemma 2.7. With that in hand, the paper would be a genuine improvement. Without it, it's a promising preprint, not a theorem.","headline":"Genuinely new operator method for zero-counting errors, with improved constants, but the key lemma's numerical verification is not rigorous and needs a certified proof before the theorem can be accepted.","tokens_in":17528,"tokens_out":3210,"would_cite":false,"duration_ms":28046,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M26","11M06","11R42"],"pacs":[],"model":"deepseek-v4-flash","headline":"A new explicit bound improves zero-counting for Dedekind zeta functions, including the Riemann zeta case.","keywords":["Dedekind zeta function","zero counting","explicit error estimates","Riemann zeta function","argument principle","nontrivial zeros","number fields","harmonic inequalities"],"falsifier":"Compute H(b,t) from Lemma 2.1 on a fine grid over |b| ≤ 1/2 and 0 < t ≤ 6 using interval arithmetic; if any sample point gives H > 0, the lemma is false. Equivalently, verify by independent high-precision computation that the degree-13 polynomial h′(t) has exactly seven real roots and that h takes negative values at the three local maxima; a fourth real root or a positive local maximum would refute the proof.","tokens_in":16486,"feed_emoji":"🔢","tokens_out":10052,"duration_ms":84785,"temperature":0.7,"pith_summary":"This paper sets out to improve the explicit error term in the asymptotic formula for N_K(T), the number of nontrivial zeros of a Dedekind zeta function ζ_K with imaginary part at most T counted with multiplicity and boundary half-weight. The proof applies the argument principle to a shifted rectangle and bounds the resulting sum over zeros using a harmonic inequality with four numerically optimized parameters. For every number field K it produces a uniform error term valid for all T ≥ 1, and specializing to K = Q gives a new explicit bound for the Riemann zeta function. If the argument is correct, these are the sharpest explicit zero-counting error terms currently available.","feed_headline":"Sharper zero-count error bound for Dedekind zeta functions","feed_subtitle":"The new estimate also improves the explicit error term for the Riemann zeta function.","key_machinery":"The engine of the proof is Lemma 2.1, which bounds the arctangent sum f(b,t,d) = 2 arctan((b+d)/t) + 2 arctan((−b+d)/t) − arctan((b+2d)/t) − arctan((−b+2d)/t) between two rational functions of t with parameters d, a1, a2, a3. The lemma asserts a harmonic function H(b,t) is never positive on |b| ≤ 1/2, t ≠ 0, for the chosen constants. The proof invokes the maximum principle and reduces the boundary check to the roots of a degree-13 polynomial, whose seven real roots are listed after a computer-assisted 'multiplying out'. This inequality is what lets the author replace the infinite zero sum by logarithmic derivatives of ξ_K at the point 1/2 + d + iT, which are then estimated by adding the cont","core_discovery":"The central claim is Theorem 1.1: for every number field K with degree n_K and discriminant d_K, and for every T ≥ 1, |N_K(T) − (T/π) log(d_K (T/(2πe))^{n_K}) − 1.919| ≤ 0.194(log d_K + n_K log T) + 5.543 n_K + 0.462. Corollary 1.2 specializes to the Riemann zeta function: |N(T) − (T/(2π)) log(T/(2πe))| ≤ 0.097 log T + 3.962 for T ≥ 1, which improves on the best previously published explicit bound. The proof starts from the argument principle for ξ_K, writes the logarithmic derivative as an absolutely convergent sum over nontrivial zeros via the Weierstrass factorization, and groups zeros with their functional-equation partners. The contribution of each pair is controlled by the inequality f","pith_inferences":["Because Lemma 2.1's numerical check is not formally certified, the theorem's validity currently rests on the reliability of that check; supplying an interval-arithmetic proof of H(b,t) ≤ 0 would convert the result into a fully rigorous theorem.","The four parameters in Lemma 2.1 were chosen by strategic testing; a systematic nonlinear optimization could find an admissible set with smaller da1, further shrinking the constant 0.194.","The same harmonic-inequality template could be applied to other L-functions in a general class, but the analogue of Lemma 2.1 would have to be proved separately for each family; this seems to be the main obstacle to transferring the method.","If combined with a verified computational framework, the method could be used to generate explicit zero-free regions or bounding boxes for zeros, with applications to prime-number estimates in number fields."],"forward_implications":["Theorem 1.1 gives uniform explicit bounds for N_K(T) for every number field K and every T ≥ 1, with constants that improve all previously known explicit results.","Corollary 1.2 yields an explicit error term for the Riemann zeta function with logarithmic coefficient 0.097, less than half of the previous best coefficient.","The paper notes that the method is in principle adaptable to any L-function with an Euler product and functional equation, provided an analogue of Lemma 2.1 can be proved.","The constants in Lemma 2.3 and Lemma 2.7 can be slightly improved by evaluating more terms in the Euler-product sum, so the numerical values are not final within the method."],"fun_headline_variants":["Tighter error term for Dedekind zeta zero counts","New bound improves Riemann zeta zero count error","Dedekind zeta zeros: improved counting error","Error term refined for Dedekind zeta function zeros"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is the claim, verified only by an unverified computer calculation, that a certain explicit two-variable function is non-positive on its whole domain. If that claim is wrong, the main theorem's error bound does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Tighter error term for Dedekind zeta zero counts","New bound improves Riemann zeta zero count error","Dedekind zeta zeros: improved counting error","Error term refined for Dedekind zeta function zeros"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000581,"raw_usage":{"total_tokens":2511,"prompt_tokens":623,"completion_tokens":1888,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":367,"completion_tokens_details":{"reasoning_tokens":1831}},"tokens_in":367,"tokens_out":1888,"duration_ms":12093,"temperature":1.0,"reasoning_tokens":1831,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T06:58:05.522163+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute H(b,t) from Lemma 2.1 on a fine grid over |b| ≤ 1/2 and 0 < t ≤ 6 using interval arithmetic; if any sample point gives H > 0, the lemma is false. Equivalently, verify by independent high-precision computation that the degree-13 polynomial h′(t) has exactly seven real roots and that h takes negative values at the three local maxima; a fourth real root or a positive local maximum would refute the proof.","supporting_citations":[],"review_version":1}