{"id":"455af4f8-07a3-4c64-a53f-2589c943d820","arxiv_id":"2505.07352","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The logarithm of the Riemann zeta function near the critical line, scaled by sqrt(log log T), converges in distribution to a complex Brownian motion.","lead":"This paper proves that the logarithm of the Riemann zeta function, sampled along horizontal lines that approach the critical line, converges statistically to a Brownian motion as the sampling range grows. The result extends Selberg's central limit theorem and yields new limiting laws for the maximum, sign changes, and local time of the zeta function.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Tightness near the critical line depends on an unproved zero-sum estimate in Proposition 4.4; the gap is fillable but must be supplied before Theorem 1.1 is fully established.","rationale":"The paper's main theorem is plausible and the finite-dimensional convergence is standard. The proof of tightness is reduced to the fourth-moment bound (2.17), and the only genuinely unproved technical input is the zero-sum estimate in Proposition 4.4. This is exactly the reader's weakest assumption, and it is load-bearing because it controls the behaviour of ζ'/ζ when σ_2 ≤ σ_c, i.e. precisely when α approaches 1 and the process should converge to the Brownian endpoint. I checked the surrounding argument: the estimate is likely provable from the zero-density estimate already cited, and the rest of the tightness proof appears coherent, so the current verdict should remain conditional pending a written proof. I also noted two minor downstream errors—the claim |ζ(σ+iτ)| ≤ 2 for σ ≥ 3/2 in Corollary 1.3 should be an O(1) bound, and Corollary 1.5 states a limit of the iterated-logarithm ratio where Brownian law gives only a limsup—but these do not affect the central theorem. The unproved zero-sum estimate is the single most serious gap, and a concrete derivation as outlined would settle whether it lands.","tokens_in":14180,"tokens_out":57418,"duration_ms":532242,"concrete_test":"Supply the missing derivation: using N(η,T) ≪ T log T exp(−η logT/4), split zeros into dyadic β-shells and, for each zero ρ=1/2+β+iγ, integrate (t−γ)^{-2} over [0,T] outside the excluded interval [γ−εx^{4β}/logT, γ+εx^{4β}/logT]. If the total contribution is O_ε(T(logT)^2), then the claimed expectation holds after dividing by T, and the enlargement of Yε can be justified by Markov's inequality. If instead the total grows faster than T(logT)^2, Proposition 4.4 fails and the tightness proof near α=1 is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of functional convergence to Brownian motion is supported by finite-dimensional convergence and a Kolmogorov tightness argument. The most load-bearing unresolved step is in the proof of Proposition 4.4, immediately after equation (4.16): the estimate E[Σ_ρ 1/(τ−γ)^2 1_{τ∉Yε}] ≪_ε (log T)^2 is asserted without proof. This estimate is the only mechanism controlling |ζ'/ζ(σ+iτ)−ζ'/ζ(σ_c+iτ)| on the high-probability event where σ_2 ≤ σ_c, i.e. for α close to 1. Without it, the bound |ζ'/ζ(σ+iτ)−ζ'/ζ(σ_c+iτ)| ≪_ε log T on a set of measure at least 1−O(ε) cannot be concluded, so the moment bound (2.15) is not established for the whole interval α∈[0,1] and the tightness argument collapses exactly at the approach to the critical line. The paper then argues that Yε can be enlarged while keeping its measure O(εT); this enlargement step also relies on the missing estimate. The assertion is plausible and likely provable from the zero-counting bound N(η,T) ≪ T log T exp(−η logT/4) already used in Lemma 4.2, but the proof is not supplied. A further minor mismatch is that the proof of Proposition 4.4 writes 'if T ≤ t ≤ 2T', whereas τ is uniform on [0,T]; this can presumably be repaired by a standard dyadic decomposition, but it is not spelled out.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a functional central limit theorem for the rescaled logarithm of the Riemann zeta function along horizontal lines approaching the critical line. For τ uniform in [0,T], the process Z^{(T)}(α) = (log ζ(1/2 + (log T)^{-α} + iτ))/√(log log T), α∈[0,1], is shown to converge in law in C_0([0,1]) to a standard complex Brownian motion. The proof proceeds via finite-dimensional convergence based on a lemma of Bourgade, and a Kolmogorov-type tightness criterion. The main technical work is the proof of moment bounds for differences of log ζ at two nearby heights, using Selberg's approximation of ζ'/ζ by Dirichlet polynomials and estimates for sums over zeta zeros. Several corollaries are derived: a reflection principle for the maximum of log|ζ|, an arcsine law, a law of the iterated logarithm, convergence of local times, and an almost-sure sign-change result.","tokens_in":14560,"tokens_out":28826,"duration_ms":243913,"significance":"The paper's main novelty is the tightness argument for the horizontal profile, which goes beyond the previously known finite-dimensional CLT. If the missing zero-sum estimate is supplied, the result establishes a natural Brownian scaling limit for the horizontal profile of log ζ, unifying Selberg's CLT with a reflection principle for the maximum. The presentation is clear, the Dirichlet-sum moment estimates are worked out in detail, and the finite-dimensional input is correctly attributed to Bourgade. The corollaries are concrete and follow from standard Brownian motion facts. The central claim is substantial and, modulo the gap discussed below, the proof strategy is sound.","major_comments":[{"comment":"The estimate E[Σ_ρ 1/(τ−γ)^2 1_{τ∉Y_ε}] ≪_ε (log T)^2, stated immediately after (4.16) and used to justify (4.17), is asserted without proof or reference. This estimate is load-bearing: it is the only mechanism that controls |ζ'/ζ(σ+iτ) − ζ'/ζ(σ_c+iτ)| on the event σ_2 ≤ σ_c, i.e., for α close to 1. Without it, the bound (4.17) and consequently the moment bound (2.15) for the critical case are not established. Please supply a proof, for example by splitting the sum over zeros and using the zero-counting bound N(η,T) from Lemma 4.2. In addition, the transition from the expectation bound to an enlarged exceptional set Y_ε with measure O(εT) on which Σ_ρ 1/(t−γ)^2 ≪_ε (log T)^2 requires an explicit argument (for instance, Markov's inequality); the current sentence 'As a result, we can increase the size of Y_ε...' is too terse for a step that is essential to the argument.","section":"Section 4.2, proof of Proposition 4.4"},{"comment":"The proof of (4.16) is written for T ≤ t ≤ 2T, whereas Theorem 1.1 takes τ uniform on [0,T]. This mismatch is not addressed in the manuscript. A standard dyadic decomposition of [0,T] would repair the argument, but the step is not spelled out. As written, the displayed derivation does not directly apply to the range of τ in the main theorem.","section":"Section 4.2, proof of Proposition 4.4"}],"minor_comments":[{"comment":"The inequality |ζ(σ+iτ)| ≤ 2 for σ ≥ 3/2 is false; for example, ζ(3/2) ≈ 2.612. The argument only requires an O(1) bound, so the corollary is unaffected, but the statement should be corrected.","section":"Corollary 1.3"},{"comment":"The displayed supremum in (1.6) is over σ∈[1/2,3/2] of log|ζ(1/2+σ+iτ)|, which corresponds to Re s ∈ [1,2]. This does not match the horizontal interval [1/2+1/logT, 3/2] covered by the process in Theorem 1.1; the intended interval is likely σ∈[0,1] (or [1/logT,1]). Please correct the notation.","section":"Corollary 1.3, equation (1.6)"},{"comment":"The bound in (4.4) contains min(x^{−η1/2} log x, (σ1−σ2)x^{−η2/2}), but a direct integration of the error bound from Lemma 4.1 gives, up to constants, min((log x)^{−1} x^{−η2/2}, (η1−η2) x^{−η2/2}). The displayed first branch appears to be a typo. This does not invalidate the subsequent estimates, since the second branch is used in the critical cases, but the formula should be corrected.","section":"Equation (4.4)"},{"comment":"In the definition of Y_ε the interval radius is ε x^{4β}/log T, but in the proof of the second part the comparison is written with log x in place of log T. This changes the implied constant in the bound β ≤ log(1/ε)/log x; the conclusion still holds after adjusting the constant in the definition of σ_c (e.g., replacing 40 by 40 log(20/ε) up to a constant). Please clarify.","section":"Lemma 4.2"},{"comment":"The proof of the tightness criterion is terse when it says 'Replacing ε by 2ε, we may replace Z by Z 1_{A_T^ε}'. For the reader's convenience, the argument should explicitly note that the probability of the complement of A_T^ε is absorbed into the small-probability term in the dyadic estimate, so the indicator in the moment assumption is sufficient. This is standard but should be stated.","section":"Theorem 2.3"}],"recommendation":"major_revision","confidential_remarks":"The paper is a serious contribution to probabilistic number theory. The main issue is the unproved zero-sum estimate in Proposition 4.4, which is load-bearing for the tightness near α=1. The estimate is plausible and likely provable from standard zero-counting bounds, so the gap appears fillable within the manuscript's scope. I recommend requesting a major revision that supplies the missing proof and clarifies the related steps. The other issues are minor and do not affect the central result."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper does one real new thing: it proves tightness for the rescaled log-zeta process on horizontal lines, turning Bourgade's finite-dimensional convergence into a functional Brownian motion limit. The author says that is the contribution, and that is an accurate self-assessment. The proof strategy is clear and mostly standard Selberg machinery, and the corollaries (reflection principle, arcsine law, local time, sign changes) are legitimate consequences if the main theorem holds. I want to give credit for the honest framing and the useful repurposing of [4].\n\nThe soft spots are real but not obviously fatal. Corollary 1.3 states |ζ(σ+iτ)|≤2 for σ≥3/2, which is false (ζ(3/2)≈2.612); an O(1) bound is enough, so the proof of the corollary survives with a one-line fix. That's minor.\n\nThe load-bearing issue is in the proof of Proposition 4.4, right after (4.16). The bound E[Σ_ρ 1/(t−γ)^2 1_{t∉Yε}] ≪_ε (log T)^2 is asserted without proof or citation. This is the only mechanism controlling the difference of log-derivatives between σ and σ_c when σ_2 ≤ σ_c, i.e. exactly the regime α near 1. Without it, the moment bound (2.15) is not established on the whole interval, and tightness breaks at the approach to the critical line. It is plausible that the estimate follows from the zero-counting bound N(η,T) already used in Lemma 4.2, but that step is not written out. A competent referee could check it, but it is not a formality.\n\nThere is also a small mismatch: the proof of Proposition 4.4 works with t in [T,2T], while τ is uniform on [0,T]. This looks repairable by a dyadic decomposition, but it is not spelled out. The extension to α∈[0,∞) is sketched rather than fully proved; that is acceptable for a remark, though a referee might want a bit more detail.\n\nOverall, I think the central argument is sound in structure and the gap is fillable. I would not cite it yet; I would wait for a version with the missing estimate written down. But it deserves a serious referee: send it to review, ask a careful probabilist who knows zero-density estimates to check Proposition 4.4, and it could be a nice paper.","headline":"The new tightness argument for a Brownian functional limit of zeta on horizontal lines is credible and largely well-executed, but one load-bearing estimate in Proposition 4.4 is asserted without proof and needs to be supplied.","tokens_in":15053,"tokens_out":2629,"would_cite":false,"duration_ms":24071,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M06","60F17","60J65"],"pacs":[],"model":"deepseek-v4-flash","headline":"As T grows, the rescaled logarithm of ζ along a one-parameter family of lines approaching the critical line converges to a standard complex Brownian motion.","keywords":["Riemann zeta function","Selberg central limit theorem","Brownian motion","functional convergence","tightness","reflection principle","arcsine law","critical line"],"falsifier":"Compute the expectation over uniform t of $\\sum_\\rho 1/(t-\\gamma)^2 1_{t\\notin Y_\\varepsilon}$ using tables of high zeros up to a large height; if for a fixed ε the quantity exceeds $C_\\varepsilon(\\log T)^2$ as T grows, the unproved estimate after equation (4.16) fails and the tightness argument near α close to 1 would not go through.","tokens_in":1826,"feed_emoji":"📈","tokens_out":2265,"duration_ms":98367,"temperature":0.7,"pith_summary":"The paper aims to prove that the whole horizontal family of rescaled log ζ values near the critical line, indexed by a parameter α that moves the line from 1/2 to 1/2 + 1/log T, converges as T grows to a single standard complex Brownian motion. This extends the classical central limit theorem for ζ from a single point to a process: not only does each vertical slice look Gaussian, but the joint behavior across nearby slices is Brownian. If correct, this explains limit laws for the maximum of log|ζ|, an arcsine law for the set where |ζ| ≥ 1, and frequent sign changes of log|ζ|. The paper identifies its main contribution as tightness of the family of random functions, with the finite-dimensional Gaussian limits following from known methods.","feed_headline":"Zeta's horizontal slices converge to Brownian motion","feed_subtitle":"Rescaled log ζ values across nearby vertical lines converge to one Brownian path, unlocking new limit laws.","key_machinery":"The argument is carried by three pieces. First, a smoothed Dirichlet-series approximation writes $\\zeta'/\\zeta(s)$ as a finite sum over prime powers with carefully chosen weights $\\Lambda_x(n)$, with $x = T^{1/20}$, plus an error term that can be controlled through zero sums. Second, a Kolmogorov-type tightness criterion reduces the problem to a fourth-moment estimate $E[|Z^{(T)}(a)-Z^{(T)}(b)|^4 1_{A_T^\\varepsilon}] \\ll_\\varepsilon |a-b|^2$. Third, the fourth moment is computed via a mean-value estimate for exponential sums and a prime-sum estimate, while an exceptional set $Y_\\varepsilon$ of measure $O(\\varepsilon T)$ removes zeta zeros that would make the approximation degenerate; a separate argument handles the region where the second point is extremely close to the critical line.","core_discovery":"The paper's central claim is Theorem 1.1: for T > 10 and τ uniform on [0,T], define $Z^{(T)}(\\alpha) = (\\log\\log T)^{-1/2}\\log\\zeta(1/2 + (\\log T)^{-\\alpha} + i\\tau)$ for $\\alpha \\in [0,1]$. Then $Z^{(T)}$ converges in law in $C_0([0,1],\\mathbb{C})$ to a standard complex Brownian motion $B = (B_1 + iB_2)/\\sqrt{2}$, where $B_1$ and $B_2$ are independent standard real Brownian motions. The finite-dimensional limits are centered Gaussian vectors with covariance $\\mathrm{Cov}(Y_i,Y_j) = 1\\wedge\\alpha_i\\wedge\\alpha_j$, which is exactly the covariance structure of Brownian motion, and the paper's new ingredient is proving the sequence of random functions is tight. A remark extends the statement to $\\alpha \\in [0,\\infty)$, where the limit stops at $B_1$ once $\\alpha \\ge 1$.","pith_inferences":["Beyond the paper: the analogous process for Haar-distributed unitary matrices, with $\\alpha \\mapsto n^{-\\alpha}$ in place of $(\\log T)^{-\\alpha}$, likely converges to the same complex Brownian motion; if so, it would yield a reflection-principle-type maximum law for characteristic polynomials.","Beyond the paper: the unproved zero-sum estimate is the only obstacle to removing the exceptional set $Y_\\varepsilon$; sharpening it would give a cleaner tightness statement with no need to cut out neighborhoods of zeta zeros.","Beyond the paper: the same Selberg-class machinery should transfer to other L-functions with Euler products, giving Brownian functional limits and the same reflection, arcsine, and sign-change corollaries for those families."],"forward_implications":["The horizontal maximum of log|ζ| obeys a reflection-principle analogue: $(\\log\\log T)^{-1/2}\\max_{\\sigma\\ge 1/2}\\log|\\zeta(\\sigma+i\\tau)|$ converges to $|N(0,1/2)|$, with upper tail probability $2\\int_u^\\infty (2\\pi)^{-1/2}e^{-x^2/2}\\,dx$.","The logarithmic measure of σ in $[1/\\log T,1]$ for which $|\\zeta(1/2+\\sigma+i\\tau)| \\ge 1$ converges to an arcsine law, $P(M_T \\le y) \\to (2/\\pi)\\arcsin\\sqrt{y}$.","The running supremum of $|\\log|\\zeta||$ over α near 0 satisfies a law of the iterated logarithm inherited from Brownian motion.","The local time of Re $Z^{(T)}$ converges weakly to Brownian local time, implying that for any fixed N, $\\log|\\zeta(1/2+\\sigma+i\\tau)|$ changes sign at least N times on $\\sigma \\in [1/2,3/2]$ with probability tending to 1.","Slices far apart in the α parameter become decorrelated with covariance $1\\wedge\\alpha_i\\wedge\\alpha_j$, matching the independent-increment structure of Brownian motion."],"supporting_citations":[{"why":"Supplies the classical central limit theorem for ζ and the smoothed Dirichlet-series decomposition of ζ'/ζ used throughout the proof.","marker":"[17]"},{"why":"Provides the finite-dimensional Gaussian covariance computation and prime-sum estimates reused in Theorem 2.1.","marker":"[4]"},{"why":"States the tightness criterion that Theorem 2.3 adapts, with the exceptional-set modification.","marker":"[15]"},{"why":"Gives the mean-value estimate for exponential sums used to compute fourth moments.","marker":"[12]"},{"why":"Supplies the reflection principle, arcsine law, and iterated-logarithm law used in the corollaries.","marker":"[14]"},{"why":"Provides the law of the iterated logarithm for random walks used in Corollary 1.5.","marker":"[10]"}],"fun_headline_variants":["Zeta's logs turn into Brownian motion","Riemann zeta's limiting shape is Brownian","ζ's scaled values become a Brownian path","Brownian motion emerges from zeta's logs","Zeta's fluctuations follow a Brownian law"],"cache_read_input_tokens":17024,"weakest_assumption_plain":"The proof rests on an unproved estimate: the sum over zeta zeros of $1/(t-\\gamma)^2$, for t outside a small exceptional set, is at most a constant times $(\\log T)^2$; if that estimate needs a larger exceptional set or a larger power of log T, the argument bounding ζ'/ζ near the critical line breaks and the main theorem is not established.","fun_headline_variants_meta":{"raw":{"variants":["Zeta's logs turn into Brownian motion","Riemann zeta's limiting shape is Brownian","ζ's scaled values become a Brownian path","Brownian motion emerges from zeta's logs","Zeta's fluctuations follow a Brownian law"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000284,"raw_usage":{"total_tokens":1651,"prompt_tokens":899,"completion_tokens":752,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":515,"completion_tokens_details":{"reasoning_tokens":680}},"tokens_in":515,"tokens_out":752,"duration_ms":7720,"temperature":1.0,"reasoning_tokens":680,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:18:48.093708+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the expectation over uniform t of $\\sum_\\rho 1/(t-\\gamma)^2 1_{t\\notin Y_\\varepsilon}$ using tables of high zeros up to a large height; if for a fixed ε the quantity exceeds $C_\\varepsilon(\\log T)^2$ as T grows, the unproved estimate after equation (4.16) fails and the tightness argument near α close to 1 would not go through.","supporting_citations":[{"cited_title":"Selberg, Contributions to the theory of the Riemann zeta-function , Arch","cited_arxiv_id":null,"evidence_quote":"Supplies the classical central limit theorem for ζ and the smoothed Dirichlet-series decomposition of ζ'/ζ used throughout the proof."},{"cited_title":"Bourgade, Mesoscopic fluctuations of the zeta zeros , Probab","cited_arxiv_id":null,"evidence_quote":"Provides the finite-dimensional Gaussian covariance computation and prime-sum estimates reused in Theorem 2.1."},{"cited_title":"Prokhorov, Convergence of random processes and limit theorems in probability theory , Teor","cited_arxiv_id":null,"evidence_quote":"States the tightness criterion that Theorem 2.3 adapts, with the exceptional-set modification."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the mean-value estimate for exponential sums used to compute fourth moments."},{"cited_title":"and Yor Revuz M., Continuous Martingales and Brownian Motion , Grundlehren der mathematischen Wissenschaften, Springer Berlin Heidelberg, 2004","cited_arxiv_id":null,"evidence_quote":"Supplies the reflection principle, arcsine law, and iterated-logarithm law used in the corollaries."},{"cited_title":"Khintchine, Uber einen Satz der Wahrscheinlichkeitsrechnung , Fundamenta Mathematicae 6 (1924), 9–20","cited_arxiv_id":null,"evidence_quote":"Provides the law of the iterated logarithm for random walks used in Corollary 1.5."}],"review_version":1}