Brownian behaviour of the Riemann zeta function around the critical line

Louis Vassaux

REVIEW 2 major objections 5 minor 18 references

Brownian behaviour of the Riemann zeta function around the critical line

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2505.07352 v2 pith:5EP2JFSZ submitted 2025-05-12 math.NT math.PR

Louis Vassaux This is my paper · ORCID

classification math.NTmath.PR MSC 11M0660F1760J65

keywords RiemannzetafunctionSelbergcentrallimittheoremBrownianmotionfunctionalconvergencetightnessreflectionprinciplearcsinelawcriticalline

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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 $\alpha \ge 1$.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

This 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].

The 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.

The 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.

There 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.

Overall, 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.

Referee Report

2 major / 5 minor

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.

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 (2)

[Section 4.2, proof of Proposition 4.4] 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 4.2, proof of Proposition 4.4] 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.

minor comments (5)

[Corollary 1.3] 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.
[Corollary 1.3, equation (1.6)] 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.
[Equation (4.4)] 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.
[Lemma 4.2] 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.
[Theorem 2.3] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: finite-dimensional convergence is imported from an external published source, and the new tightness argument is an independent moment calculation.

full rationale

The paper's main theorem is derived from two independent ingredients. First, convergence of finite-dimensional distributions is obtained by applying Bourgade's Lemma 2.2 [4], a published external result on random Dirichlet polynomials, after verifying its hypotheses. This is not a self-citation, and the cited lemma does not assume the Brownian functional limit; the paper explicitly acknowledges that this part repurposes the methods of [4]. Second, tightness is proved from scratch via a Kolmogorov–Prokhorov criterion with A=4, B=2. The required fourth-moment bound is decomposed into a Dirichlet-polynomial part, proved using Lemmas 3.2–3.4, and an error-term part, handled with Selberg's classical identity and zero-density bounds from Selberg [17]. No parameter is fitted to the target distribution, and no conclusion of Theorem 1.1 is assumed in the proof. The corollaries (reflection principle, arcsine law, iterated logarithm law, local time, sign changes) are standard Brownian-motion facts transferred by the functional convergence; the choice of the measure µ_T in Corollary 1.4 is a change of variables matching the parametrization σ = 1/(log T)^α, which is a legitimate time-change rather than a definition of the target. The only load-bearing statement asserted without proof is the zero-sum bound E[Σ_ρ (t−γ)^{-2} 1_{t∉Yε}] ≪_ε (log T)^2 in the proof of Proposition 4.4; this is an unproved analytic estimate and a possible correctness gap, but it is not an input of the theorem restated as an output. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no free parameters and no invented entities. It relies on standard analytic number theory results including Bourgade's Dirichlet-series criterion, Selberg's zero-density estimates and decomposition, and the Montgomery-Vaughan mean value theorem. One estimate in the proof of Proposition 4.4, a bound on a sum over zero reciprocals outside a rare set, is asserted without proof or citation; this is the main unverified input.

assumptions (6)

domain assumption Lemma 2.2 (Bourgade [4]): convergence criterion for Dirichlet series with slowly varying coefficients.
Used in Section 2.1 to derive finite-dimensional convergence. It is proven in Bourgade 2010 and accepted as background.
domain assumption Zero-density estimate N(η,T) ≪ T log T exp(-c η log T) for nontrivial zeroes with β ≥ η, 0 ≤ γ ≤ T.
Invoked in Lemma 4.2 to control the exceptional set Y_ε and in the proof of Proposition 4.4. Standard Selberg-type estimate, quoted from [17].
domain assumption Selberg's decomposition (2.18): ζ'/ζ(s) = -Σ_{n≤x^3} Λ_x(n)/n^s + e_x(s), with bounds for e_x from Lemma 4.1.
Basis of the moment computation in Section 4.1; taken from Selberg [17].
domain assumption Uniform bound |ζ(σ+iτ)| = O(1) for σ ≥ 3/2 (paper states the stronger but false |ζ| ≤ 2).
Used in Corollary 1.3 to discard the region σ ≥ 3/2. The true O(1) bound is classical, but the paper's ≤2 statement is incorrect.
ad hoc to paper Unproved estimate E[Σ_ρ 1/(t-γ)^2 1_{t∉Yε}] ≪_ε (log T)^2, used to justify (4.17) in Proposition 4.4.
Stated without proof in Section 4.2; not clearly a standard result, and the proof relies on it to control the difference of ζ'/ζ near σ_c.
standard math Montgomery-Vaughan mean value theorem (Lemma 3.4) and Kolmogorov-Prokhorov tightness criterion (Theorem 2.3) are standard or proved in-paper.
Used in Lemma 3.2 and Section 2.2 respectively.

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Cite this review

Pith. "Pith review of Brownian behaviour of the Riemann zeta function around the critical line." pith.science (2026). https://pith.science/paper/5EP2JFSZ

@misc{pith2026250507352,
  author       = {Pith},
  title        = {Pith review of: Brownian behaviour of the Riemann zeta function around the critical line},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5EP2JFSZ}},
  note         = {Machine review of arXiv:2505.07352}
}

abstract

We establish a Brownian extension to Selberg's central limit theorem for the Riemann zeta function. This implies various limiting distributions for $\zeta$, including an analogue of the reflection principle for the maximum of the Brownian motion: as $T$ diverges, for any $u>0$ we have \[ \frac{1}{T}\cdot {\rm meas}\Big\{0\leq t\leq T:\max_{\sigma\geq \tfrac{1}{2}}\log|\zeta(\sigma+i t)|\geq u \sqrt{\tfrac{1}{2}\log \log T} \Big\}\to 2 \displaystyle\int_u^{\infty} \frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi}}\mathrm{d} x. \]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 15 canonical work pages

[4]

Bourgade, Mesoscopic fluctuations of the zeta zeros , Probab

P. Bourgade, Mesoscopic fluctuations of the zeta zeros , Probab. Theory Related Fields 148 (2010), no. 3-4, 479–500

work page 2010
[1]

Arguin, D

L.-P. Arguin, D. Belius, P. Bourgade, M. Radziwi l l, and K. Soundararajan, Maximum of the Riemann zeta function on a short interval of the critical line , Comm. Pure Appl. Math. 72 (2019), no. 3, 500–535

work page 2019
[2]

Arguin, P

L.-P. Arguin, P. Bourgade, and M. Radziwi l l,The Fyodorov-Hiary-Keating Conjecture. I, arXiv preprint arXiv:2007.00988 (2020)

arXiv 2020
[3]

II, arXiv preprint arXiv:2307.00982 (2023)

, The Fyodorov-Hiary-Keating Conjecture. II, arXiv preprint arXiv:2307.00982 (2023)

arXiv 2023
[5]

Bourgade and J

P. Bourgade and J. Keating, Quantum chaos, random matrix theory, and the Riemann ζ-function, Chaos, Prog. Math. Phys., vol. 66, Birkh¨ auser/Springer, Basel, 2013. 13

work page 2013
[6]

Fyodorov, G

Y. Fyodorov, G. Hiary, and J. Keating, Freezing Transition, Characteristic Polynomials of Random Matrices, and the Riemann Zeta Function, Physical Review Letters 108 (2012)

work page 2012
[7]

Fyodorov and J

Y. Fyodorov and J. Keating, Freezing transitions and extreme values: random matrix theory, and disordered landscapes , Philosophical Transactions of the Royal Society A 372 (2014), no. 2007, 20120503

work page 2014
[8]

Harper, The Riemann zeta function in short intervals [after Najnudel, and Arguin, Belius, Bourgade, Radziwi l land Soundararajan], Ast´ erisque422, S´ eminaire Bourbaki

A. Harper, The Riemann zeta function in short intervals [after Najnudel, and Arguin, Belius, Bourgade, Radziwi l land Soundararajan], Ast´ erisque422, S´ eminaire Bourbaki. Vol. 2018/2019. Expos´ es 1151–1165(2020), Exp. No. 1161, 391–414

work page 2020

Show all 18 references

[9]

Harper, On the partition function of the Riemann zeta function, and the Fyodorov-Hiary-Keating conjecture , arXiv preprint arXiv:1906.05783 (2019)

A. Harper, On the partition function of the Riemann zeta function, and the Fyodorov-Hiary-Keating conjecture , arXiv preprint arXiv:1906.05783 (2019)

2019 arXiv
[10]

Khintchine, Uber einen Satz der Wahrscheinlichkeitsrechnung , Fundamenta Mathematicae 6 (1924), 9–20

A. Khintchine, Uber einen Satz der Wahrscheinlichkeitsrechnung , Fundamenta Mathematicae 6 (1924), 9–20

1924
[11]

H. L. Montgomery, The pair correlation of zeros of the zeta function , Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), Proc. Sympos. Pure Math., Vol. XXIV, Amer. Math. Soc., Providence, RI, 1973, pp. 181–193

1972
[12]

H. L. Montgomery and R. C. Vaughan, Hilbert’s inequality, J. London Math. Soc. (2) 8 (1974), 73–82

1974
[13]

Najnudel, On the extreme values of the Riemann zeta function on random intervals of the critical line , Probab

J. Najnudel, On the extreme values of the Riemann zeta function on random intervals of the critical line , Probab. Theory Related Fields 172 (2018), no. 1-2, 387–452

2018
[14]

and Yor Revuz M., Continuous Martingales and Brownian Motion , Grundlehren der mathematischen Wissenschaften, Springer Berlin Heidelberg, 2004

D. and Yor Revuz M., Continuous Martingales and Brownian Motion , Grundlehren der mathematischen Wissenschaften, Springer Berlin Heidelberg, 2004

2004
[15]

Prokhorov, Convergence of random processes and limit theorems in probability theory , Teor

Y. Prokhorov, Convergence of random processes and limit theorems in probability theory , Teor. Veroyatnost. i Primenen. 1 (1956), 177–238 (Russian, with English summary)

1956
[16]

Radziwi l l and K

M. Radziwi l l and K. Soundararajan,Selberg’s central limit theorem for log|ζ(1/2 +it)|, Enseign. Math. 63 (2017), no. 1-2, 1–19

2017
[17]

Selberg, Contributions to the theory of the Riemann zeta-function , Arch

A. Selberg, Contributions to the theory of the Riemann zeta-function , Arch. Math. Naturvid. 48 (1946), no. 5, 89–155

1946
[18]

Soundararajan, The distribution of values of zeta and L-functions, ICM—International Congress of Mathematicians

K. Soundararajan, The distribution of values of zeta and L-functions, ICM—International Congress of Mathematicians. Vol. 2. Plenary lectures, EMS Press, Berlin, [2023] ©2023, pp. 1260–1310. 14

2023

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [4]

Bourgade, Mesoscopic fluctuations of the zeta zeros , Probab

P. Bourgade, Mesoscopic fluctuations of the zeta zeros , Probab. Theory Related Fields 148 (2010), no. 3-4, 479–500

work page 2010

[2] [1]

Arguin, D

L.-P. Arguin, D. Belius, P. Bourgade, M. Radziwi l l, and K. Soundararajan, Maximum of the Riemann zeta function on a short interval of the critical line , Comm. Pure Appl. Math. 72 (2019), no. 3, 500–535

work page 2019

[3] [2]

Arguin, P

L.-P. Arguin, P. Bourgade, and M. Radziwi l l,The Fyodorov-Hiary-Keating Conjecture. I, arXiv preprint arXiv:2007.00988 (2020)

arXiv 2020

[4] [3]

II, arXiv preprint arXiv:2307.00982 (2023)

, The Fyodorov-Hiary-Keating Conjecture. II, arXiv preprint arXiv:2307.00982 (2023)

arXiv 2023

[5] [5]

Bourgade and J

P. Bourgade and J. Keating, Quantum chaos, random matrix theory, and the Riemann ζ-function, Chaos, Prog. Math. Phys., vol. 66, Birkh¨ auser/Springer, Basel, 2013. 13

work page 2013

[6] [6]

Fyodorov, G

Y. Fyodorov, G. Hiary, and J. Keating, Freezing Transition, Characteristic Polynomials of Random Matrices, and the Riemann Zeta Function, Physical Review Letters 108 (2012)

work page 2012

[7] [7]

Fyodorov and J

Y. Fyodorov and J. Keating, Freezing transitions and extreme values: random matrix theory, and disordered landscapes , Philosophical Transactions of the Royal Society A 372 (2014), no. 2007, 20120503

work page 2014

[8] [8]

Harper, The Riemann zeta function in short intervals [after Najnudel, and Arguin, Belius, Bourgade, Radziwi l land Soundararajan], Ast´ erisque422, S´ eminaire Bourbaki

A. Harper, The Riemann zeta function in short intervals [after Najnudel, and Arguin, Belius, Bourgade, Radziwi l land Soundararajan], Ast´ erisque422, S´ eminaire Bourbaki. Vol. 2018/2019. Expos´ es 1151–1165(2020), Exp. No. 1161, 391–414

work page 2020