{"id":"af04a311-cc7b-4f11-943d-5e4d21c7e36d","arxiv_id":"2505.23238","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":6,"one_line_summary":"A regulated area-integral convergence criterion is claimed to prove RH, but the proof's key divergence estimate is contradicted by the paper's own excision of zero neighborhoods.","lead":"A paper claims to prove the Riemann Hypothesis by showing that a regulated surface integral over the critical strip converges exactly when all non-trivial zeta zeros lie on the critical line. The proof fails because the claimed divergence for an off-line zero is computed inside the disks that the integral's own definition removes.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The necessity direction integrates over the disk that Definition 3.5.3 removes: with the correct epsilon_n cutoff the local contribution is finite, so divergence under an off-line zero is not established.","rationale":"The reader's verdict is correct, and the identified weak point is load-bearing. The central equivalence requires both that an off-line zero forces divergence of W(R) and that W(R) converges unconditionally. The first direction is contradicted by the paper's own Definition 3.5.3: the divergent radial integral is taken over the removed disk B(rho, epsilon), whereas the integration domain starts at epsilon_n > 0, making the local contribution finite; the 1/(2R) factor even sends a fixed zero's contribution to zero. The second direction is also unsupported, since the Section 5.4 estimate depends on m_R, which shrinks as more zero disks with radii epsilon_n -> 0 enter the domain; the asserted R-independent bound does not follow. These are internal consistency failures, not disagreements with the surrounding literature. Standard facts quoted in Chapter 2 are not at issue. The recommendation is unchanged: REJECT.","tokens_in":28444,"tokens_out":8073,"duration_ms":93848,"concrete_test":"Recompute Lemma 4.3.3 using the domain required by Definition 3.5.3: for an off-line zero rho of index n and R > |Im rho|, replace the integral from 0 to epsilon of r^(1-lambda*m) dr by the integral from epsilon_n to epsilon of r^(1-lambda*m) dr and apply the 1/(2R) normalization. If the result is finite and tends to 0 as R -> infinity, the divergence lemma and the necessity direction collapse.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 4.3.3 (and its variants 4.3.4-4.3.8) defines I_rho(R) as the integral over B(rho, epsilon) and concludes divergence from the integral of r^(1-lambda*m) dr from 0 to epsilon being infinite. But W(R) integrates over Dreg_R = D_R minus the union of K_n and K_pol, where K_n = B(rho_n, epsilon_n) and epsilon_n = (n+N0)^(-alpha). Once R > |Im rho|, an off-line zero rho is one of the removed centers, and the actual local contribution is (1/(2R)) times the integral from epsilon_n to epsilon of r^(1-lambda*m) dr, times bounded angular factors. This is finite for every fixed R and tends to 0 after the 1/(2R) normalization. Thus the inequality W(R) >= I_rho(R) = infinity is false, and the necessity direction of Satz 6.2.1/6.4.1 has no basis. The same cut-off confusion appears in Section 5.4: the bound W(R) <= m_R^(-lambda) times the integral of |sigma-1/2|^(-p) is called R-independent, but m_R = min over Dreg_R of |zeta| tends to 0 as epsilon_n -> 0, so the convergence proof is also not uniform in R.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript claims to prove the Riemann Hypothesis by constructing a regulated, normalized surface integral W(R) over a vertical strip in the critical strip, with integrand JC(s) = 1/(|zeta(s)|^lambda |Re(s)-1/2|^p), and by excising small disks around the zeros and the pole. The central theorem is stated as lim_{R->infty} W(R) < infinity if and only if the Riemann Hypothesis holds. The proof proceeds in two directions: a necessity direction claiming that any zero off the critical line makes the integral diverge, and a sufficiency direction claiming that W(R) converges without assuming RH, using the alternating Dirichlet series or a regulated Mellin representation. The paper concludes that both directions imply the Riemann Hypothesis.","tokens_in":28770,"tokens_out":4684,"duration_ms":47604,"significance":"If the central equivalence were correct, this would be a major breakthrough in analytic number theory, reducing RH to a convergence criterion for an explicit integral and avoiding all known difficult aspects of the problem. The manuscript is clearly organized, states definitions and lemmas in a formal style, and references standard works. However, the claimed proof rests on load-bearing internal inconsistencies: the necessity direction integrates over disks that the construction explicitly removes, and the convergence proof uses a lower bound m_R that depends on R and degenerates as R grows. These are not merely presentation issues; they invalidate both directions of the proposed equivalence. The paper also contains a later section that fits constants to known zeros, which is empirical rather than a derivation. For these reasons, the significance of the claimed result cannot be recognized in the current form.","major_comments":[{"comment":"The divergence claim for an off-line zero rho computes the local contribution I_rho(R) as (1/(2R)) times the integral of r^{1-lambda m} from r = 0 to epsilon, i.e., over the full disk B(rho, epsilon). This contradicts Definition 3.5.3, which removes the disk B(rho_n, epsilon_n) from D_reg^R, with epsilon_n = (n+N0)^{-alpha}. For the actual domain the radial integral starts at r = epsilon_n and equals (epsilon^{2-lambda m} - epsilon_n^{2-lambda m})/(2-lambda m), which is finite for each fixed R and tends to 0 after division by 2R. Hence the inequality W(R) >= I_rho(R) = infinity is false, and the necessity direction of Satz 6.2.1 has no basis.","section":"§5.4.2, Lemma 5.4.1 and Satz 5.4.2"},{"comment":"The convergence proof bounds W(R) by m_R^{-lambda} times the finite integral of |sigma - 1/2|^{-p}, and then claims that all estimates are independent of R. This is incorrect because m_R = min_{s in D_reg^R} |zeta(s)| depends on R: as R increases, more zeros enter the strip and the excision radii epsilon_n tend to 0, so m_R tends to 0. The factor m_R^{-lambda} therefore grows without bound, and the presented estimate does not imply limsup_{R->infty} W(R) < infinity. The same defect appears in §5.4.3, where the analogous bound is used.","section":"§4.3.2, Lemma 4.3.2"},{"comment":"In the claimed proof that W(R) -> 0 under RH, the contribution from S2(R) is bounded by (1/(2R)) * C R^{alpha p} * Area(S2(R)) <= (1/(2R)) * C R^{alpha p} * 4R = 2 C R^{alpha p}, which diverges for any alpha > 1 and p > 0. The text states that alpha is chosen so that the term is controlled, but no such choice exists because alpha p > 0. Moreover, the claim that JC is bounded by a constant M on S1(R) is not uniform in R, since near the boundaries of the excised disks |zeta|^{-lambda} grows like epsilon_n^{-lambda m}. Thus the convergence statement for the RH case is not proved.","section":"§5.2 vs §6.4.1"},{"comment":"Theorem 5.2.1 states the equivalence with lim_{R->infty} W(R) = 0, while the abstract and Satz 6.4.1 use lim_{R->infty} W(R) < infinity. These are different statements: the proof of Lemma 4.3.2 (if correct) would give convergence to 0 under RH, whereas Section 5.4 claims only finiteness. The manuscript does not reconcile these formulations, and the formal main theorem is therefore ambiguous.","section":"§7.1.5"},{"comment":"The 'asymptotic approximation' for the zero ordinates gamma_n contains constants a, b, c, d that are numerically fitted to the known zeros of the zeta function. This is empirical curve fitting, not a derivation from the integral model, and it does not provide independent support for the main theorem. In addition, the vertical projection Phi(t) in §7.1.1 is introduced under the assumption of RH, so the reconstruction argument is conditional on the statement being proved.","section":"§5.4.2"}],"minor_comments":[{"comment":"In Satz 3.6.3 the local integrals are computed over r from epsilon to delta and are finite, but the concluding sentence says that the local integrals around the zeros diverge 'grundsätzlich'. This contradicts the preceding calculation; the divergence is removed by the excision, and the exposition should be corrected to avoid confusion.","section":"§4.3.1"},{"comment":"The summary of the case distinction says 'konvergiert exakt dann gegen null', while the stated criterion is lim W(R) < infinity. These are not the same, and the inconsistency should be resolved.","section":"Definition 3.5.1"},{"comment":"The symbol delta is used both for the half-width of the vertical strip in Definition 3.5.1 and for the deviation beta - 1/2 in Lemma 4.3.3. This overloaded notation makes the local asymptotic formulas harder to follow.","section":"§4.1"},{"comment":"There is a typo 'Definitoin' in the sentence introducing epsilon_n, and the phrase 'alle singulären Beiträge ... vollständig reguliert' is unclear because the excision is a regularization of the domain, not a deletion of the contributions.","section":"§7.1.5"},{"comment":"The approximation formula has a typo in the phrase 'für kleineren' and the displayed formula contains an undefined parameter 'n' and a term 'd/n^7' whose origin is not explained. The figures referenced as Abbildung 7.1 and 7.2 are not included in the manuscript text.","section":"Bibliography"}],"recommendation":"reject","confidential_remarks":"The manuscript is posted in math.GM and is submitted as a thesis at a university. The core proof has a fundamental domain/integration mismatch that cannot be repaired by local revision: the necessity direction integrates over excised disks, and the convergence proof relies on an R-dependent lower bound. I see no salvage within the manuscript's scope; a fully rewritten argument would be needed. I also note that Section 7.1.5, which presents fitted constants as a 'streng analytische Herleitung', would require a separate and much more careful treatment before it could be considered a derived formula."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a genuine, well-organized attempt at an integral criterion for RH, but the central equivalence is not proved. The necessity direction integrates over the very disks the regulated domain cuts out; the convergence direction needs a uniform lower bound that the construction cannot supply. I agree with the stress-test note.\n\nWhat the paper does well: the object JC(s) is explicit, the local singularity analysis around zeros is correct, and the case split in Chapter 4 is thorough. The regularization idea—excise disks with summable radii—is coherent, and the references are standard. The author is plainly serious and the exposition is readable.\n\nThe problems. Lemma 4.3.3 and its siblings compute the local contribution of an off-line zero as (1/2R)∫_0^ε r^{1−λm} dr and call it infinite. But by Definitions 3.5.3 and 3.5.4, the point ρ is removed with radius ε_n=(n+N0)^{−α}. The actual local integral starts at r=ε_n; it is finite for each fixed R and, after the 1/(2R) normalization, goes to 0. So the inequality W(R) ≥ ∞ is false, and the necessity direction of Theorem 6.4.1 has no basis. This is an internal contradiction with the paper's own domain definition, not a subtle analytic gap.\n\nThe convergence direction has the same illness in a different form. Section 5.4 bounds |ζ|^{−λ} by m_R^{−λ}, where m_R is the minimum on the regulated domain, and then treats the estimate as R-independent. As R grows, new zeros enter, the excised disks shrink, and m_R tends to 0; the bound does not survive passage to the limit. A fixed-width strip away from zeros would prove only local regularity, not global convergence.\n\nChapter 7 is not evidence for the main theorem. The constants a, b, c, d in the asymptotic zero formula are explicitly fitted to known zeros; presenting that as a derivation from the integral model is not support. It undercuts the claim of a purely analytic, parameter-free proof.\n\nThe citation pattern is standard and the organization is clear. But both directions of the claimed equivalence rest on the same kind of mistake: treating the excised singularities as still present in the integral. I would not send this to peer review; it is a desk reject. If an editor wants to be generous, a two-page note to the author pointing at Lemma 4.3.3 and the m_R issue would be more useful than a full referee cycle.","headline":"A serious but invalid RH proof attempt: both directions of the claimed equivalence fail because the excised zero disks are quietly reinserted in the estimates.","tokens_in":29287,"tokens_out":3216,"would_cite":false,"duration_ms":36834,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M26","11M06"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims to prove the Riemann Hypothesis by constructing a weighted area integral $W(R)$ whose convergence is equivalent to every nontrivial zero lying on the critical line $\\Re(s)=1/2$.","keywords":["Riemann hypothesis","zeta zeros","critical line","area integral","regulated integral","analytic criterion","Dirichlet series","Mellin representation"],"falsifier":"Build a model function with exactly one simple zero $\\rho$ off the critical line and integrate the same regulated integrand over the same strip, removing a disk of radius $\\varepsilon_n=(n+N_0)^{-\\alpha}$ around $\\rho$ with the paper's parameters $\\lambda=2$, $p=\\tfrac12$. The local annulus contribution is proportional to $\\int_{\\varepsilon_n}^{\\varepsilon} r^{1-\\lambda}\\,dr$, which is finite for every $\\varepsilon_n>0$, so a finite limit, or a divergence rate different from the paper's prediction, would directly contradict the claimed equivalence.","tokens_in":28178,"feed_emoji":"🧮","tokens_out":12467,"duration_ms":120809,"temperature":0.7,"pith_summary":"This paper tries to prove the Riemann Hypothesis by tying it to the convergence of a weighted area integral over the critical strip. The integral $W(R)$ averages $|\\zeta(s)|^{-\\lambda}|\\Re(s)-\\tfrac12|^{-p}$ over a strip from which small disks around every zero and around the pole at $s=1$ have been removed. The central claim is that $W(R)$ has a finite limit as $R\\to\\infty$ if and only if every nontrivial zero of $\\zeta$ lies on the critical line $\\Re(s)=\\tfrac12$. Having argued that any off-line zero forces divergence and that $W(R)$ actually converges, the paper concludes that the Riemann Hypothesis is true. If the argument is right, RH would follow from a single analytic convergence statement rather than from numerical or spectral input.","feed_headline":"Area-integral criterion puts all zeta zeros on the critical line","feed_subtitle":"The paper makes the integral's convergence equivalent to RH, then asserts that convergence holds.","key_machinery":"The central object is the regulated normalized area integral $W(R)=\\frac{1}{2R}\\int\\int_{D_R^{\\mathrm{reg}}} |\\zeta(s)|^{-\\lambda}|\\Re(s)-\\tfrac12|^{-p}\\,dA(s)$. The domain $D_R^{\\mathrm{reg}}$ is the vertical strip $\\{0\\le\\Re(s)\\le1,\\,|\\Im(s)|\\le R\\}$ with disks $B(\\rho_n,\\varepsilon_n)$ around each nontrivial zero and a disk around the pole $s=1$ removed; the excision radii are $\\varepsilon_n=(n+N_0)^{-\\alpha}$ with $\\alpha>1$, chosen so the total removed area is finite. The exponents $\\lambda\\ge2$ and $0<p<1$ make $|\\zeta|^{-\\lambda}$ dominate the horizontal weight $|\\Re(s)-\\tfrac12|^{-p}$, so a zero off the critical line becomes a potential divergence source. The proof then runs on two halves: a case analysis showing that any off-line zero makes $W(R)$ diverge, and a convergence proof for $W(R)$ built on exact representations of $\\zeta$, whose combination forces all zeros onto the critical line.","core_discovery":"On the paper's own terms, the discovery is an equivalence: $\\lim_{R\\to\\infty} W(R)<\\infty$ holds exactly when all nontrivial zeros of $\\zeta(s)$ satisfy $\\Re(s)=\\tfrac12$. The construction is deliberately singularity-sensitive: near a zero $\\rho$ of multiplicity $m$, $|\\zeta(s)|$ behaves like $|s-\\rho|^m$, so the integrand behaves like $|s-\\rho|^{-\\lambda m}$ times a horizontal weight, and with $\\lambda\\ge2$ this is non-integrable at an off-line zero if the integration actually reaches the zero. Chapter 4 classifies all zero configurations and asserts that every deviation from the critical line therefore forces divergence; Chapter 5 claims to prove convergence of $W(R)$ using only the alternating Dirichlet series and a regulated Mellin representation, without knowing zero locations. The formal conclusion in Section 6.4 is that the Riemann Hypothesis is true.","pith_inferences":["A natural stress test is to apply the same construction to a meromorphic function with one prescribed off-line zero and summable excision radii; if the regulated limit stays finite, the necessity direction would need additional hypotheses not stated in the paper.","If the convergence proof in Chapter 5 genuinely uses only exact representations of $\\zeta$, the same integrand could provide a concrete analytic criterion for Dirichlet $L$-functions or other $L$-functions with the same zero symmetries, a transfer the paper only sketches.","The vertical projection $\\Phi(t)$ introduced in Chapter 7 suggests a numerical probe: tracking whether $W(R)$ stabilizes for large $R$ in high-precision computation would be consistent with the paper's conclusion, although it would not by itself prove the limit statement.","The paper's equivalence, if correct, would place RH in the family of integral-convergence criteria, potentially connecting to existing zero-counting and explicit-formula methods without requiring the zeros to be known in advance."],"forward_implications":["If the central claim is correct, all nontrivial zeros of $\\zeta(s)$ lie on $\\Re(s)=\\tfrac12$, so the Riemann Hypothesis follows as a corollary.","The equivalence turns RH into a single analytic convergence check, with no need to locate individual zeros; the excision radii depend only on the zero index.","The method is claimed to transfer to other zeta and $L$-functions, giving analogous convergence criteria for their zero distributions.","Because the normalization by $1/(2R)$ cancels the strip's linear area growth, a finite limit detects structural zero misplacement rather than the trivial growth of the domain.","Any hypothetical off-line zero would force $W(R)$ to diverge, so the criterion would expose the smallest deviation from the critical line."],"supporting_citations":[{"why":"Supplies the analytic continuation, the alternating Dirichlet series, the zero-free region, and the asymptotic zero-counting law that the convergence analysis and case distinction rely on.","marker":"[7]"},{"why":"Introduces the zeta function and the statement that all nontrivial zeros lie on the critical line, the target of the paper's proof.","marker":"[6]"},{"why":"Provides the Dirichlet-series and Euler-product foundations used to fix where $\\zeta$ is holomorphic and nonvanishing.","marker":"[1]"},{"why":"Gives the functional equation and the four-fold zero symmetry that the case analysis uses to cover all configurations.","marker":"[2]"},{"why":"Provides the zero-free region and analytic estimates used to control $\\zeta$ away from its zeros in the critical strip.","marker":"[5]"},{"why":"Underpins the measure-theoretic integrability statements that let the integrand be treated as locally $L^1$ after excision.","marker":"[4]"}],"fun_headline_variants":["Integral convergence equals all zeta zeros on critical line","Riemann Hypothesis proven by singularity-sensitive integral","Paper: integral regulates zeta zeros onto critical line","Zeta zeros forced to critical line by convergence criterion","Riemann Hypothesis shown via regulated normalized integral"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof that one off-line zero makes the integral diverge assumes the integral is taken down to that zero, even though the regulated domain removes a positive-radius disk around every zero; without a separate argument that the removed disks still accumulate divergence, the necessity direction is not established.","fun_headline_variants_meta":{"raw":{"variants":["Integral convergence equals all zeta zeros on critical line","Riemann Hypothesis proven by singularity-sensitive integral","Paper: integral regulates zeta zeros onto critical line","Zeta zeros forced to critical line by convergence criterion","Riemann Hypothesis shown via regulated normalized integral"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000234,"raw_usage":{"total_tokens":1427,"prompt_tokens":809,"completion_tokens":618,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":425,"completion_tokens_details":{"reasoning_tokens":545}},"tokens_in":425,"tokens_out":618,"duration_ms":6838,"temperature":1.0,"reasoning_tokens":545,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:50:21.309844+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Build a model function with exactly one simple zero $\\rho$ off the critical line and integrate the same regulated integrand over the same strip, removing a disk of radius $\\varepsilon_n=(n+N_0)^{-\\alpha}$ around $\\rho$ with the paper's parameters $\\lambda=2$, $p=\\tfrac12$. The local annulus contribution is proportional to $\\int_{\\varepsilon_n}^{\\varepsilon} r^{1-\\lambda}\\,dr$, which is finite for every $\\varepsilon_n>0$, so a finite limit, or a divergence rate different from the paper's prediction, would directly contradict the claimed equivalence.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the analytic continuation, the alternating Dirichlet series, the zero-free region, and the asymptotic zero-counting law that the convergence analysis and case distinction rely on."},{"cited_title":"The method ismethodologically independentof spectral, statistical, or numerical approaches and does not require knowledge of the exact zeros","cited_arxiv_id":null,"evidence_quote":"Introduces the zeta function and the statement that all nontrivial zeros lie on the critical line, the target of the paper's proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Dirichlet-series and Euler-product foundations used to fix where $\\zeta$ is holomorphic and nonvanishing."},{"cited_title":"Die Methode istmethodisch unabhängigvon spektralen, statistischen oder nume- rischen Ansätzen und benötigt keine Kenntnis der exakten Nullstellen","cited_arxiv_id":null,"evidence_quote":"Gives the functional equation and the four-fold zero symmetry that the case analysis uses to cover all configurations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the zero-free region and analytic estimates used to control $\\zeta$ away from its zeros in the critical strip."},{"cited_title":"Beweis der Riemannschen Vermutung \\\"uber ein reguliertes normiertes Integralmodell","cited_arxiv_id":"2505.23238","evidence_quote":"Underpins the measure-theoretic integrability statements that let the integrand be treated as locally $L^1$ after excision."}],"review_version":1}