{"id":"1c728271-a388-42f3-aca3-3b51bd06e86a","arxiv_id":"2507.04150","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The joint moments of log ζ and a smoothed zero-counting function factorize into Gaussian moments, showing asymptotic independence.","lead":"This paper proves that the logarithm of the Riemann zeta function and the local statistics of its zeros behave like independent Gaussian variables when measured together. It combines Selberg's central limit theorem with Hughes and Rudnick's mock-Gaussian results, and under the Riemann Hypothesis it can handle a wider class of zero statistics.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4.2's combinatorial enumeration is the most load-bearing unproven step; a miscount in the ten-relation decomposition would change the main constant in Theorem 1.1.","rationale":"The reader's conditional verdict is reasonable, but the stated weakest assumption (RH) is not the most load-bearing issue for the paper's main unconditional theorem. The genuinely central and least secure step is the combinatorial evaluation in Proposition 4.2, which converts the diagonal sum into the exact Gaussian constant. I could not find a concrete error in the argument; the proof is very technical and the missing details are likely routine. However, the step is asserted rather than demonstrated, and since the main theorem's leading coefficient is exact, this is where a hidden mistake would most likely live. A targeted small-case enumeration would settle whether the concern lands. Thus the verdict should remain conditional on filling this gap, matching the reader's recommendation without changing it.","tokens_in":16562,"tokens_out":44991,"duration_ms":491217,"concrete_test":"Independently verify the combinatorial core of Proposition 4.2 for small parameters. For h=ℓ=2, k=4, and a generic even φ with sufficiently small Fourier support, write a short script that enumerates all ordered tuples (p1,p2,q1,q2,n1,n2,n3,n4) with p_i,q_i ≤ x primes and n_j either a prime or a square of a prime up to T^η, satisfying the diagonal equality after removing the off-diagonal error. Compute the exact coefficient of (log log T)^2 in the resulting finite sum and compare it with 2! · μ_4 · σ_φ^4 = 6σ_φ^4. Also test whether every enumerated solution can be decomposed into the ten relations (4.9)–(4.10); if any solution is not covered, the classification is incomplete and Proposition 4.2 needs revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central unconditional result, Theorem 1.1, depends on converting the diagonal sum in Proposition 4.1 into the exact Gaussian constant μ_k σ_φ^k h! (log log T)^h. That conversion is done in Proposition 4.2, and specifically in the step after equation (4.8): the paper states that each solution of the equality ∏ p_i ∏_{j∈J} n_j = ∏ q_i ∏_{j∈Jc} n_j 'has to be in one of the relations' (4.9)–(4.10), that the left-hand side 'factors as a product' of the individual sums, and that the number of ways to pair the p_i and q_j 'is simply h!'. These are assertions, not proofs. The constants C(u,J) that count combinations of relations are never computed; the proof only analyzes the special case u1=h, u4=k/2, and dismisses all other configurations with vague bounds. A missed relation type (for example, a relation involving one left prime, one right prime, and one n_j on either side) or an overcount caused by repeated primes would alter the leading coefficient, because the main term is obtained by exact counting, not by an upper bound. This is not a claim of error; the step is likely routine. But as written it is the least secure link between the Dirichlet-polynomial moments and the advertised Gaussian moments, and it is more central to the paper's main theorem than the Riemann Hypothesis, which only affects the conditional Theorem 1.2.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the joint distribution of log ζ(1/2+it) and the linear statistics N_φ(t) of the nontrivial zeros of ζ. The main results are an unconditional mixed-moment asymptotic (Theorem 1.1) for a power of N_φ−φ̂(0) times powers of log ζ and its conjugate, with Fourier support η < 2/(k+2); an RH-conditional version (Theorem 1.2) for ℑ log ζ with the larger support η < 2/k; and an RH-conditional correlation estimate (Proposition 1.5) showing that the first mixed moment of log ζ and N_φ−φ̂(0) is −φ(0)/2 + O(√log log T / log T). The proof follows the standard Selberg–Tsang route: an explicit formula for N_φ, approximation of log ζ by a short Dirichlet polynomial, reduction to diagonal moments, and a combinatorial evaluation of the diagonal sum. The paper also derives weighted central limit theorems (Corollaries 1.3 and 1.4) as consequences of the moment estimates.","tokens_in":16838,"tokens_out":40694,"duration_ms":408676,"significance":"If the proofs are correct, this is a substantial contribution: it establishes a genuine joint version of Selberg's CLT and the Hughes–Rudnick mock-Gaussian moment results, with explicit constants and no free parameters. The unconditional range η < 2/(k+2) is natural given the Hölder-based approximation, and the RH-conditional extension for ℑ log ζ reaches the conjecturally sharp barrier η < 2/k. The correlation estimate of Proposition 1.5 is a nice concrete illustration of the slow decay of independence. The methods are standard but the combination is new. The main theorems are clearly stated and the paper is well organized. However, as detailed below, a load-bearing step in the proof of the central Theorem 1.1 is asserted rather than proved, so the paper needs revision before it is fully convincing.","major_comments":[{"comment":"The proof of Proposition 4.2 asserts that every solution of the diagonal equality ∏ p_i ∏_{j∈J} n_j = ∏ q_i ∏_{j∈Jc} n_j 'has to be in one of the relations' (4.9)–(4.10), and that the left-hand side of (4.8) 'factors as a product' of the corresponding sums. This step directly produces the leading constant μ_k σ_φ^k h! in Theorem 1.1, yet it is not proved. In particular, the constants C(u,J) are introduced but never computed or bounded; the exclusion of all configurations with u1 < h or with u7+u10 > 0 is only sketched; and the claim that the number of p–q matchings is 'simply h!' requires qualification when primes repeat, because the same tuple can then arise from several bijections. A rigorous enumeration is needed, including a verification that all configurations outside u1 = h = ℓ and u4 = k/2 contribute at most O((log log T)^((h+ℓ−1)/2)). Since Theorem 1.2 and Corollaries 1.3–1.4 inherit this step, the gap is load-bearing.","section":"§4, Proposition 4.2, after Eq. (4.8)"},{"comment":"The bounds for the individual sums S1,...,S10 are stated with 'one verifies', and the subsequent counting argument is informal. The paper does not write down the linear relations between h, ℓ, k and the multiplicities u1,...,u10; such an accounting is necessary to justify the assertions that u2 = u3 = u5 = u6 = u8 = u9 = 0 when h = ℓ = u1, and that u7 = u10 = 0 unless the contribution is absorbed into the error term. Without these details, the proof of the main constant in Proposition 4.2 is not fully verifiable. Please provide a complete count of the relation types and a proof that the only leading-order configuration is the one with all p's paired to q's and all n's paired among themselves.","section":"§4, Eq. (4.11) and subsequent paragraph"}],"minor_comments":[{"comment":"The letter h is used both for the number of p-variables and for the nonzero integer difference in the diagonal approximation; this notation clash is confusing. Please use a different symbol (e.g., m) for the integer difference.","section":"§4, Eqs. (4.5)–(4.6)"},{"comment":"The bound for E2 is described only as a 'routine calculation' after an application of Hölder's inequality; please spell out the mixed-moment estimate for ℜ E_x and ℑ E_x, especially the role of the (k+2)-th moment of S*_φ in fixing the support condition η < 2/(k+2).","section":"§3, Proposition 3.1"},{"comment":"The estimates for the weighted moments of A3 and R are stated without proof ('by a similar calculation'). Since these bounds are needed for Theorem 1.2, a few explanatory lines would make the argument easier to check.","section":"§3, Proposition 3.2"},{"comment":"The error term after applying Goldston's formula (2.5) should be O((log T / T) ∑_{n≤T^η} Λ(n)), not O((1/T) ∑ Λ(n)); the displayed bound is too optimistic by a factor log T. The final estimate is still acceptable, but the displayed line should be corrected.","section":"§2, proof of Proposition 1.5, Eq. (2.6)"},{"comment":"For k = 0 the support condition on φ̂ is unnecessary, since S*_φ does not appear in the moment; stating the k = 0 case separately (or noting that the condition is vacuous for that case) would avoid an artificial restriction.","section":"§1, Theorem 1.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely correct and the main theorems are valuable. The principal issue is the unproved combinatorial enumeration in Proposition 4.2, which is load-bearing for Theorem 1.1 and hence for the corollaries. I would be willing to accept after the authors supply a complete proof of that step and tighten the sketched estimates in Section 3. The manuscript fits the journal's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does something real: it proves a joint moment formula for log ζ and the linear statistics of zeros, giving asymptotic independence (in the sense of moments) between the prime contribution and the zero contribution. Theorem 1.1 is new, and Theorem 1.2 sharpens the support under RH in a way that matches the Hughes–Rudnick barrier for the imaginary part. The correlation estimate in Proposition 1.5 is also a nice touch, with the slow decay clearly isolated under RH. The structure is clean and the unconditional versus conditional results are properly separated.\n\nThe proof follows the expected route: explicit formula, Dirichlet polynomial approximation, Selberg–Tsang moment bounds, and then a diagonal computation. The soft spot, as the stress-test note says, is Proposition 4.2. The step after (4.8) asserts that every solution of the equality must fall into one of ten relations, that the sum factors as a product over those relations, and that the pairing count is simply h!. None of those assertions is given a real proof. The constants C(u,J) are called for but never computed except in the case u1=h, u4=k/2. Since the Gaussian constant in the main theorem comes from exact counting, not an upper bound, this is load-bearing. I do not think the step is wrong; it looks routine and the authors' description is plausible. But as written, a referee cannot verify the leading constant without redoing the combinatorics. That is a fixable gap, not a fatal flaw.\n\nA minor related issue: the sums S2, S3 and S_i for i≥5 are dismissed with ‘one verifies’, and S4's evaluation is also terse. These are probably fine, but they contribute to the sense that the hard combinatorial part is being glossed.\n\nThe citation pattern is fine; the authors cite their own earlier work where it is contextual, and the load-bearing external theorems (Selberg–Tsang, Hughes–Rudnick, Goldston) are clearly identified. The paper is honest about the RH dependence and the support restriction coming from Hölder.\n\nFor whom: analytic number theorists working on value distribution of zeta, zero statistics, and hybrid models. It deserves a serious referee; the main theorem is interesting enough to spend referee time on, with the expectation of a request for a more detailed combinatorial section.","headline":"Main theorem is new and valuable, but the proof of Proposition 4.2 skips the exact combinatorics that carries the main constant; likely routine, yet currently the least secure load-bearing step.","tokens_in":17361,"tokens_out":1022,"would_cite":true,"duration_ms":14125,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M06","11M26"],"pacs":[],"model":"deepseek-v4-flash","headline":"At leading order the logarithm of zeta and the counting statistics of its zeros are asymptotically independent Gaussians, so weighting by nearby zero density leaves Selberg's central limit theorem intact.","keywords":["Selberg's central limit theorem","Riemann zeta function","linear statistics of zeros","mock-Gaussian behavior","method of moments","Riemann Hypothesis","one-level density","critical line"],"falsifier":"Compute the $h=\\ell=1$, $k=2$ mixed moment $\\frac{1}{T}\\int_T^{2T}|\\log\\zeta(1/2+it)|^2(N_\\varphi(t)-\\hat\\varphi(0))^2\\,dt$ for a fixed $\\varphi$ with $\\operatorname{supp}\\hat\\varphi\\subset(-1/2,1/2)$, and divide by $\\log\\log T$; the theorem forces this ratio to converge to $\\sigma_\\varphi^2$, so a numerical evaluation at large $T$ showing the ratio drifting away from $\\sigma_\\varphi^2$ would refute the claimed asymptotic independence.","tokens_in":16357,"feed_emoji":"","tokens_out":15925,"duration_ms":157644,"temperature":0.7,"pith_summary":"This paper establishes a joint central limit theorem: the logarithm of the Riemann zeta function on the critical line and the linear statistics of its nontrivial zeros are asymptotically independent Gaussian objects at leading order. For fixed $h,\\ell$ and even $k$, whenever the test function $\\varphi$ has Fourier transform supported in $(-2/(k+2),\\,2/(k+2))$, the mixed moment $\\frac{1}{T}\\int_T^{2T}(\\log\\zeta(1/2+it))^h(\\overline{\\log\\zeta(1/2+it)})^\\ell(N_\\varphi(t)-\\hat\\varphi(0))^k\\,dt$ equals $\\mathbf{1}(h=\\ell)\\,\\mu_k\\sigma_\\varphi^k\\,h!(\\log\\log T)^h$ plus a smaller error term, which is exactly the factorization predicted by a complex Gaussian paired with a real Gaussian of variance $\\sigma_\\varphi^2$. Under the Riemann Hypothesis the support for the imaginary part extends to the natural barrier $(-2/k,\\,2/k)$, and the same Gaussian factorization holds. The paper also shows that the correlation between $\\log\\zeta$ and the centered zero statistic is not zero at the next order: under RH it is asymptotically $-\\varphi(0)/2$, so the correlation coefficient decays only like $1/\\sqrt{\\log\\log T}$. This matters because it says Selberg's Gaussian value distribution survives even when one conditions on the local density of nearby zeros, while the approach to independence is genuinely slow.","feed_headline":"Zeta's log stays Gaussian under zero-statistic weights","feed_subtitle":"A joint moment proof shows the logarithm of zeta and the density of nearby zeros become asymptotically independent Gaussian variables","key_machinery":"The argument runs on a weighted moment calculation with three ingredients. Lemma 2.1 is an explicit formula that rewrites the centered zero statistic as the prime-power sum $S_\\varphi(t)=-(2/\\log T)\\sum_n \\Lambda(n)/\\sqrt{n}\\,\\hat\\varphi(\\log n/\\log T)\\cos(t\\log n)$ up to an error $O(1/\\log T)$, so the zero statistic becomes a Dirichlet polynomial whose coefficients are supported on primes and prime squares. The second ingredient is the classical Dirichlet polynomial $P_x(t)=\\sum_{p\\le x}p^{-1/2-it}$, which approximates $\\log\\zeta(1/2+it)$ in mean square; Propositions 3.1 and 3.2 reduce the mixed moments of $\\log\\zeta$ and $N_\\varphi$ to moments of $P_x$ and $S_\\varphi^*$, with the Riemann Hypothesis needed only for the wider support in the imaginary part. The load-bearing combinatorial step is Proposition 4.2: after expanding the product and integrating, only the diagonal terms survive, and the prime-power equalities split into ten possible pairing relations, of which only two contribute at leading order: $P_x$ pairing with itself, contributing $h!(\\log\\log T)^h$, and $S_\\varphi^*$ pairing with itself, contributing $\\mu_k\\sigma_\\varphi^k$ through the variance integral $\\sigma_\\varphi^2=\\int\\min(|u|,1)\\,\\hat\\varphi(u)^2\\,du$. The factorization into these two dominant sums is exactly what produces asymptotic independence.","core_discovery":"On its own terms, the paper's central claim is Theorem 1.1: for fixed $h,\\ell,k$ with $k$ even, and for any even real-valued $\\varphi$ with smooth compactly supported Fourier transform satisfying $\\operatorname{supp}\\hat\\varphi\\subseteq(-2/(k+2),\\,2/(k+2))$, the joint moment $\\frac{1}{T}\\int_T^{2T}(\\log\\zeta(1/2+it))^h(\\overline{\\log\\zeta(1/2+it)})^\\ell(N_\\varphi(t)-\\hat\\varphi(0))^k\\,dt$ equals $\\mathbf{1}(h=\\ell)\\,\\mu_k\\sigma_\\varphi^k\\,h!(\\log\\log T)^h$ plus an error $O((\\log\\log T)^{(h+\\ell-1)/2})$. This is the moment sequence of a standard complex Gaussian multiplied by the moment sequence of a real Gaussian with variance $\\sigma_\\varphi^2=\\int\\min(|u|,1)\\,\\hat\\varphi(u)^2\\,du$, so the two random variables are independent at leading order. Passing from moments to distributions gives Corollary 1.3: if $\\tau$ is sampled with probability weight $|N_\\varphi(\\tau)-\\hat\\varphi(0)|^k$, then $\\log\\zeta(1/2+i\\tau)/\\sqrt{\\log\\log T}$ still converges to a standard complex Gaussian. Under the Riemann Hypothesis, Theorem 1.2 achieves the same factorization for the imaginary part with the full support $\\eta<2/k$; the imaginary part, normalized by $\\sqrt{(1/2)\\log\\log T}$, converges to a standard real Gaussian under the weighted measure. Finally, Proposition 1.5 identifies the next-order correlation: under RH, $\\frac{1}{T}\\int_T^{2T}\\log\\zeta(1/2+it)(N_\\varphi(t)-\\hat\\varphi(0))\\,dt=-\\varphi(0)/2+O(\\sqrt{\\log\\log T}/\\log T)$, so the correlation coefficient is asymptotic to $-\\varphi(0)/(2\\sigma_\\varphi\\sqrt{\\log\\log T})$.","pith_inferences":["The same mechanism suggests a robust principle: any fixed polynomial in short-range zero statistics should fail to shift the leading Gaussian fluctuations of $\\log\\zeta$, so the factorization should persist for weighted measures built from several one-level densities.","Because the slow correlation decay matches the hybrid prime-zero product picture in which the zero product contributes only at second order, one would expect the $-\\varphi(0)/2$ term to be visible in short-interval averages too; computing the covariance on intervals of length $T^\\theta$ rather than $[T,2T]$ is a concrete way to test how the correlation develops.","If the Riemann Hypothesis is false, Theorem 1.2 and Proposition 1.5 lose their footing; a natural test is whether a much weaker zero-density hypothesis, rather than the full Riemann Hypothesis, is enough to push the imaginary-part support to $\\eta<2/k$."],"forward_implications":["If the theorem is right, choosing $t$ with probability proportional to $|N_\\varphi(t)-\\hat\\varphi(0)|^k$ does not change the limiting Gaussian distribution of $\\log\\zeta(1/2+it)$; Selberg's central limit theorem survives this conditioning.","Under the Riemann Hypothesis, the same holds for $\\Im\\log\\zeta$ with Fourier support up to the natural barrier $\\eta<2/k$, so the imaginary part has a real Gaussian limit under the weighted measure.","The leading mixed moments of $\\log\\zeta$ and $N_\\varphi$ factor into the product of their individual Gaussian and mock-Gaussian moments, meaning the two are asymptotically independent at the scale of Selberg's theorem.","The correlation estimate in Proposition 1.5 shows the independence is only asymptotic: the correlation coefficient between $\\log\\zeta$ and $N_\\varphi-\\hat\\varphi(0)$ is of size $1/\\sqrt{\\log\\log T}$, so the approach to independence is slow.","Setting $h=\\ell=0$ in the moment computation recovers the mock-Gaussian moment result for the one-level density with the sharp cutoff $\\omega=\\mathbf{1}_{[0,1]}$, so the weighted theorem contains that result as a special case."],"supporting_citations":[{"why":"Supplies the mock-Gaussian moment theorem for the one-level density $N_\\varphi$ and the variance $\\sigma_\\varphi^2$ that the joint moments are matched against.","marker":"[14]"},{"why":"Provides the classical Dirichlet polynomial approximation and moment estimates used to replace $\\log\\zeta$ by $P_x$ in the weighted moments.","marker":"[32]"},{"why":"Selberg's original central limit theorem for $\\log\\zeta$, the statement being extended to weighted measures.","marker":"[27]"},{"why":"Gives the hybrid Euler-Hadamard product picture separating prime and zero contributions, which motivates the independence claim.","marker":"[13]"},{"why":"Provides the effective mean-value estimate for $\\int\\log\\zeta\\, n^{\\pm it}$ used to compute the correlation in Proposition 1.5.","marker":"[12]"}],"fun_headline_variants":["Weighted zeta log keeps Gaussian law under zero stats","Zeta log and zero density become independent Gaussians","Selberg's CLT survives weighting by zeta zeros","RH extends support for weighted log zeta CLT","Zeta log's Gaussian law resists zero-statistic weighting"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is the Riemann Hypothesis, used for the full support $\\eta<2/k$ in the imaginary-part theorem and for the correlation formula $-\\varphi(0)/2$; without it, the unconditional theorem survives only with the narrower support $\\eta<2/(k+2)$.","fun_headline_variants_meta":{"raw":{"variants":["Weighted zeta log keeps Gaussian law under zero stats","Zeta log and zero density become independent Gaussians","Selberg's CLT survives weighting by zeta zeros","RH extends support for weighted log zeta CLT","Zeta log's Gaussian law resists zero-statistic weighting"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000609,"raw_usage":{"total_tokens":2925,"prompt_tokens":1122,"completion_tokens":1803,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":738,"completion_tokens_details":{"reasoning_tokens":1723}},"tokens_in":738,"tokens_out":1803,"duration_ms":14667,"temperature":1.0,"reasoning_tokens":1723,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:54:13.320517+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the $h=\\ell=1$, $k=2$ mixed moment $\\frac{1}{T}\\int_T^{2T}|\\log\\zeta(1/2+it)|^2(N_\\varphi(t)-\\hat\\varphi(0))^2\\,dt$ for a fixed $\\varphi$ with $\\operatorname{supp}\\hat\\varphi\\subset(-1/2,1/2)$, and divide by $\\log\\log T$; the theorem forces this ratio to converge to $\\sigma_\\varphi^2$, so a numerical evaluation at large $T$ showing the ratio drifting away from $\\sigma_\\varphi^2$ would refute the claimed asymptotic independence.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the mock-Gaussian moment theorem for the one-level density $N_\\varphi$ and the variance $\\sigma_\\varphi^2$ that the joint moments are matched against."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the classical Dirichlet polynomial approximation and moment estimates used to replace $\\log\\zeta$ by $P_x$ in the weighted moments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Selberg's original central limit theorem for $\\log\\zeta$, the statement being extended to weighted measures."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the hybrid Euler-Hadamard product picture separating prime and zero contributions, which motivates the independence claim."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the effective mean-value estimate for $\\int\\log\\zeta\\, n^{\\pm it}$ used to compute the correlation in Proposition 1.5."}],"review_version":1}