Pith. sign in

REVIEW 3 major objections 5 minor 2 cited by

An Analytic Zeta Function Ramp at the Black Hole Thouless Time

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The log-spectrum SFF is exactly the Riemann zeta ramp, with slope one and an O(1) Thouless time.

desk verdict Solid analytic derivation of the zeta ramp, but the O(1) Thouless time rests on an unproven identification of the infinite-zeta ramp with the finite-N truncated spectrum. read the letter →

arxiv 2505.00528 v1 pith:7FKCTGS4 submitted 2025-05-01 hep-th

Pallab Basu , Suman Das , Chethan Krishnan This is my paper · ORCID
classification hep-th
keywords spectralformfactorRiemannzetafunctionlogarithmicspectrumThoulesstimeblackholenormalmodesL-functionsdip-ramp-plateauPoissonresummation
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

This paper proves that the spectral form factor (SFF) of the deterministic spectrum $E_n = \log n$, with the early-time classical dip removed, is exactly the Riemann zeta function, and that its time-averaged ramp is linear with slope one: $\langle |\zeta(it)|^2 \rangle = \frac{\pi}{24} t$. The proof runs through the Euler-Maclaurin decomposition of the truncated partition function together with the functional equation $\zeta(s) = \chi(s)\zeta(1-s)$, whose gamma factors supply the growth $t^{1-2\beta}$ for $0 \le \beta < 1/2$. The same reasoning fixes the onset of the ramp, giving a Thouless time of $\mathcal{O}(1)$ that is independent of the truncation size $N$, the behaviour expected of black-hole microstates. The authors extend the mechanism to $L$-functions, where the ramp exponent is twice the degree times the real part of the critical line, and they derive the ramp slope $1/(1-\alpha)$ for $E_n = n^\alpha$ via Poisson resummation. A reader should care because it shows that deterministic spectra, not only random matrices, can reproduce the black-hole spectral signatures from a first-principles calculation.

What carries the argument

The load-bearing object is the Euler-Maclaurin split $Z_N(s) \approx N^{1-s}/(1-s) + \zeta(s) + \frac12 N^{-s} - \frac{s}{12}N^{-1-s}$ (Eq. (3.9)), which separates the classical dip contribution $N^{1-s}/(1-s)$ from the quantum ramp $\zeta(s)$. The second engine is the reflection property of the functional equation $\zeta(s)=\chi(s)\zeta(1-s)$: Stirling's approximation gives $|\chi(\beta+it)|\approx (t/2\pi)^{1/2-\beta}$, converting a mean-value theorem for $\Re(s)>1/2$ into the ramp $t^{1-2\beta}$ for $\beta<1/2$. For $L$-functions, the same gamma-ratio reflection yields the general ramp exponent $2d\,\mathrm{Re}(\text{critical line})$, where $d$ is the degree of the $L$-function and the critical line is the symmetry axis of the functional equation, located at real part $(w+1)/2$. Poisson resummation with a saddle-point evaluation supplies the slope $1/(1-\alpha)$ for power-law spectra $E_n=n^\alpha$.

What would settle it

Compute $I(T)=\int_1^T|\zeta(it)|^2\,dt$ at very large $T$ with controlled numerical error; if $I(T)/T^2$ does not approach $\pi/24\approx0.1309$, or if the ramp onset extracted from the truncated SFF after subtracting the classical dip visibly moves with $N$, the central claim fails.

Watch

Extended reading notes

Core claim

On its own terms, the central discovery is that Riemann's analytic continuation is the quantum correction to a truncated log spectrum, so the zeta function is the full ramp after removal of the dip. Concretely, the paper establishes Eq. (3.34): for $\beta=0$, $\langle |\zeta(it)|^2\rangle = (\pi/24)t$, so the SFF ramp has slope exactly 1 on a log-log plot, with the coefficient fixed rather than fitted. For $0\le\beta<1/2$ the ramp grows as $t^{1-2\beta}$ up to constants; at $\beta=1/2$ it grows logarithmically; and for $\beta>1/2$ it saturates to the plateau $\zeta(2\beta)$. The $s=1$ pole is read as a Hagedorn transition, and the onset of the zeta ramp, read off from its intersection with the classical dip, gives a Thouless time that is $\mathcal{O}(1)$ and $N$-independent, matching the black-hole expectation and distinguishing the log spectrum from other black-hole toy models.

Load-bearing premise

The argument assumes that the zeta term in Eq. (3.9) is the full ramp after the dip is removed, so cross terms between $N^{1-s}/(1-s)$ and $\zeta(s)$ must be negligible in the ramp window and the onset time read from $\zeta(it)\zeta(-it)$ must equal the onset for the actual truncated spectrum; neither is proved.

Editorial extensions

If this is right

  • The SFF of the $\log n$ spectrum has an exact linear ramp of slope 1 with coefficient $\pi/24$; no ensemble averaging is needed because the ramp is a deterministic property of the arithmetic sequence.
  • The log spectrum has an $\mathcal{O}(1)$, $N$-independent Thouless time; this is the first black-hole-inspired toy model, apart from random matrices, for which that has been demonstrated (the SYK model scales as $\mathcal{O}(\log N)$ and fuzzball constructions as $\mathcal{O}(N^{\#})$).
  • For $L$-functions, the nontruncated SFF at $\beta=0$ ramps forever as $t^{2d\,\mathrm{Re}(\text{critical line})}$, and truncating the Dirichlet series restores a plateau; the functional equation, not the zeros, controls the ramp.
  • For power-law spectra $E_n=n^\alpha$ with $0<\alpha<1$, the ramp slope $1/(1-\alpha)$ follows analytically from a saddle-point estimate of Poisson-resummed terms, confirming the earlier numerical guess.

Reading between the lines

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

  • One can test the $\mathcal{O}(1)$ onset independently by constructing the connected, unfolded SFF of the truncated log spectrum with a standard filtering method rather than reading the intersection with the classical dip; if the two onsets disagree, the claimed Thouless time is definition-dependent.
  • The gamma-reflection mechanism suggests a sharper criterion than the paper proves: any Selberg-class Dirichlet series with real spectral parameters should show a ramp with exponent fixed by degree and critical line, so checking a non-self-dual or higher-degree $L$-function with known data would either extend or test the claimed universality.
  • Because the ramp is deterministic, the fluctuations around $\pi t/24$ are arithmetic rather than stochastic; one could ask whether their time-averaged moments obey random-matrix universality, which the paper leaves open.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies the spectral form factor (SFF) of the logarithmic spectrum E_n = log n. It uses the Euler–Maclaurin decomposition Z_N(s) = N^{1-s}/(1-s) + ζ(s) + ... to separate a classical dip from quantum corrections, derives via the functional equation and mean-value theorems that the time-averaged quantity ⟨|ζ(it)|^2⟩ behaves as (π/24)t, and from this claims that the Riemann zeta function ramp has slope exactly 1. The authors then identify the RZF as the 'full ramp after removal of the dip', read an O(1) Thouless time from the first intersection of ζ(it)ζ(-it) with the ramp line, and argue that this matches black-hole expectations. They generalize the ramp exponent to L-functions, obtaining exponent 2d(w+1)/2 at β=0, and use Poisson resummation to derive the slope 1/(1-α) for power-law spectra E_n = n^α.

Significance. The derivation of the RZF ramp, Eqs. (3.23)-(3.34), is a clean and mostly sound application of standard mean-value theorems and Stirling approximation, and the numerical fits in Figure 6 support the constant π/24. The L-function scaling formula and the Poisson-resummation treatment of n^α spectra are concrete, falsifiable statements and give useful analytic control where previous work was numerical. If the identification of the RZF ramp with the finite-N truncated log-spectrum ramp were established, the O(1) Thouless time would be a significant and surprising result for black-hole toy models. At present, however, that identification is asserted rather than proved, and several of the paper's strongest claims depend on it.

major comments (3)
  1. [3.3, 4; Eq. (3.9)] The identification of ζ(it)ζ(-it) with the ramp of the finite-N truncated log spectrum is not established. Expanding |Z_N(it)|^2 via Eq. (3.9) produces, in addition to |ζ(it)|^2, the cross term 2 Re[N^{1-it} ζ(-it)/(1-it)]; its envelope is of order N |ζ(it)|/t ~ N t^{-1/2}, which exceeds |ζ(it)|^2 ~ t until t ≳ N^{2/3}. No estimate or cancellation argument for these cross terms is given, and Figure 3, which compares raw curves, does not prove that the RZF ramp is the ramp of the truncated SFF. Because the O(1), N-independent Thouless time in Section 4 is read directly from ζ(it)ζ(-it), the central physical claim rests on this unproven identification.
  2. [4; Fig. 8] The O(1) Thouless time is read in a regime where the asymptotic approximations used to derive the ramp are not controlled. Eq. (3.25) is a large-|t| Stirling approximation and Eq. (3.30) is a large-T mean-value estimate; neither controls the pointwise behavior of ζ(it)ζ(-it) at t = O(1). The first intersection in Figure 8 therefore lies in the transient regime and is not a derived onset time. In addition, the yellow line in Figure 8 is (π/12)t, whereas Eq. (3.34) gives the cumulative time average (π/24)T; the factor-of-two discrepancy and the distinction between a local moving average and a cumulative average need to be clarified, since the first-intersection time depends on which curve is used.
  3. [3.3, 4] The definition of the RZF as the 'full ramp after removal of the dip' makes the N-independence of the ramp, and hence the O(1) onset, partly true by construction. A statement about the actual truncated spectrum requires an independent check of the subtracted quantity |Z_N - Z_cl|^2 or of the finite-N SFF after dip removal; Figures 1-3 compare raw curves and do not isolate the claimed quantum contribution. Without such a check, the O(1) Thouless time is a property of the definition rather than a demonstrated property of the log spectrum.
minor comments (5)
  1. [3.5.4; Eq. (3.33)] Equation (3.33) writes I(T) with a lim_{T→∞} symbol in front of an expression that still depends on T; this should read I(T) = ∫_1^T ζ(it)ζ(-it) dt, and Eq. (3.34) should distinguish the cumulative average ⟨g⟩_T from the local moving-average ramp.
  2. [5; Eq. (5.14)] The expression 't^{2d w+1/2}' following Eq. (5.14) is ambiguous; it should read t^{2d(w+1)/2}.
  3. [3.5.4; Eq. (3.30)] For 1-β > 1/2 the error term in the mean-value estimate is not O(t^{2β}); a more careful statement would be O(t^{2β-1} log t) (or O(1) at β=0). The leading term is unaffected, but the formula as written is inaccurate.
  4. [8] The heading of the acknowledgments section contains a typo: 'Acknolwedgments' should be 'Acknowledgments'.
  5. [Figure 4 caption] The caption states that for β < 1/2 the function displays a persistent ramp proportional to t^{1-2β}; this is the local-average behavior, whereas Eq. (3.32) is a cumulative integral. The caption should state this distinction to avoid confusion.

Circularity Check

1 steps flagged · score 6.0 of 10

The RZF slope calculation is independent, but the O(1) Thouless-time claim is read off from ζ(it)ζ(-it), which the paper itself defines as the 'full ramp after removal of the dip'; the onset is O(1) by construction rather than derived from the finite-N truncated spectrum.

  1. self definitional [Abstract and Sec. 4 (with Sec. 3.3)]
    "This perspective yields a precise definition of RZF as the “full ramp after removal of the dip”, and allows an unambiguous determination of the Thouless time. ... It is a trivial matter to determine the “beginning” time of the full ramp from the plots of the mod square RZF as well our analytic average ramp function – we do that in Figure 8. It should also be evident from (say Figure 3 ) that the RZF ramp is indeed capturing the ramp that was “hidden under the dip”. It is immediate that the Thouless time is unambiguously O(1) from Figure 8. It is also naturally independent of N."

    In Eq. (3.9), Z_N(s) ≈ N^{1−s}/(1−s) + ζ(s) + …, so the truncated SFF is |Z_N(it)|^2, not |ζ(it)|^2; it contains cross terms such as 2 Re[N^{1−it}ζ(−it)/(1−it)] whose envelope is ∼ N|ζ(it)|/t and can exceed the |ζ|^2 ramp unless t ≳ N^{2/3}. The paper does not bound these cross terms; instead it declares ζ(s) to be the quantum contribution, i.e. the “full ramp after removal of the dip”, and then determines the Thouless time from ζ(it)ζ(−it) alone (Figure 8). The O(1), N-independent onset is therefore a property of the defining object—the untruncated zeta SFF—rather than a derived property of the full finite-N log spectrum. The headline black-hole O(1) claim is built in by this definition.

full rationale

The analytic ramp-slope calculation itself is not circular: Eq. (3.34), ⟨ζ(it)ζ(−it)⟩ = (π/24)t, follows from external mean-value theorems for |ζ(1−β+it)|^2 together with the functional equation (3.23)–(3.25), and the L-function ramp exponents similarly follow from functional equations plus Stirling approximations (E.1). These are parameter-free mathematical inputs, not fitted to the paper's own conclusions. The circularity is confined to the Thouless-time identification: the paper defines the RZF as the “full ramp after removal of the dip”, and then reads the ramp onset from ζ(it)ζ(−it), so the resulting O(1) and N-independent Thouless time is a consequence of that definition rather than of a controlled analysis of the truncated SFF |Z_N(it)|^2. Because the slope derivation retains independent content while the central black-hole O(1) claim partially reduces by construction, the appropriate score is 6.

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

The central zeta derivation is built on classical theorems (functional equation, mean-value theorems, Stirling asymptotics) and is not fitted to data. The main interpretive burden is the identification of ζ(s) as the quantum contribution after removing the dip, and the physical assumption that black hole normal modes are logarithmic. No free parameters are introduced into the derivation; numerical constants in figures are confirmatory fits, not inputs.

assumptions (6)
  • standard math Mean value theorems for |ζ(σ+it)|^2 for σ>1/2 and the log growth at σ=1/2.
    Used in Sections 3.5.1-3.5.3 to obtain plateau height ζ(2β) and log growth at β=1/2; cited to Titchmarsh [25].
  • standard math Riemann zeta functional equation ζ(s)=χ(s)ζ(1-s) with Stirling approximation |χ(β+it)|=(t/2π)^(1/2−β).
    Central to Case IV, Eqs. (3.23)-(3.25); standard textbook result.
  • domain assumption General L-functions satisfy the completed functional equation (5.7) with real spectral parameters, and L(1−w/2−β−it) has O(1) mean size.
    Used in Section 5.3.2 to derive g0(0,t)~t^(2d(w+1)/2); known for the arithmetic examples but conjectural for the full Selberg class.
  • standard math Euler-Maclaurin expansion (3.8) and the separation of Z_N(s) into the classical integral N^(1−s)/(1−s) plus quantum corrections led by ζ(s).
    Underlies Sections 3.2-3.3 and the definition of the RZF as the full ramp; the expansion itself is standard, while the classical/quantum split is interpretive.
  • domain assumption Black hole normal modes are approximately logarithmic, ω_J ~ a + b log J.
    Connects the log spectrum to black holes in Sections 1 and A.4; taken from earlier papers [7,8,12], not derived here.
  • standard math Poisson resummation identity and saddle point evaluation of the Fourier integrals.
    Used in Section 6 to derive ramp slope 1/(1−α) for E_n=n^α.

how reviews work

0 comments
Cite this review

Pith. "Pith review of An Analytic Zeta Function Ramp at the Black Hole Thouless Time." pith.science (2026). https://pith.science/paper/7FKCTGS4

@misc{pith2026250500528,
  author       = {Pith},
  title        = {Pith review of: An Analytic Zeta Function Ramp at the Black Hole Thouless Time},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7FKCTGS4}},
  note         = {Machine review of arXiv:2505.00528}
}
abstract

Black hole normal modes have intriguing connections to logarithmic spectra, and the spectral form factor (SFF) of $E_n = \log n$ is the mod square of the Riemann zeta function (RZF). In this paper, we first provide an analytic understanding of the dip-ramp-plateau structure of RZF and show that the ramp at $\beta \equiv \Re(s)=0$ has a slope precisely equal to 1. The $s=1$ pole of RZF can be viewed as due to a Hagedorn transition in this setting, and Riemann's analytic continuation to $\Re(s)< 1$ provides the quantum contribution to the truncated $\log n$ partition function. This perspective yields a precise definition of RZF as the ''full ramp after removal of the dip'', and allows an unambiguous determination of the Thouless time. For black hole microstates, the Thouless time is expected to be $\mathcal{O}(1)$--remarkably, the RZF also exhibits this behavior. To our knowledge, this is the first black hole-inspired toy model that has a demonstrably $\mathcal{O}(1)$ Thouless time. In contrast, it is $\mathcal{O}(\log N)$ in the SYK model and expected to be $\mathcal{O}(N^{\#})$ in supergravity fuzzballs. We trace the origins of the ramp to a certain reflection property of the functional equation satisfied by RZF, and suggest that it is a general feature of $L$-functions--we find evidence for ramps in large classes of $L$-functions. As an aside, we also provide an analytic determination of the slopes of (non-linear) ramps that arise in power law spectra using Poisson resummation techniques.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Analytic Spread Complexity from Level Statistics: From Chaos to Integrability

    hep-th 2026-08 conditional novelty 7.0 of 10

    The finite-time peak of spread complexity is controlled by the Fourier transform of the nearest-neighbour energy-level spacing distribution.

  2. Virasoro Blocks and Trouble at the Euclidean Horizon

    hep-th 2025-05 conditional novelty 6.0 of 10

    A geodesic Witten diagram built directly in Euclidean BTZ coordinates reproduces the semi-classical Virasoro block and links the half-thermal-period timescale to the probe geodesic reaching the horizon.

Reference graph

Works this paper leans on

36 extracted references · 15 canonical work pages · cited by 2 Pith papers

  1. [1]

    Vishveshwara,Scattering of Gravitational Radiation by a Schwarzschild Black-hole, Nature 227 (1970) 936

    C.V. Vishveshwara,Scattering of Gravitational Radiation by a Schwarzschild Black-hole, Nature 227 (1970) 936

    1970

  2. [2]

    ’t Hooft,On the Quantum Structure of a Black Hole, Nucl

    G. ’t Hooft,On the Quantum Structure of a Black Hole, Nucl. Phys. B256 (1985) 727

    1985

  3. [3]

    S. Das, C. Krishnan, A.P. Kumar and A. Kundu,Synthetic fuzzballs: a linear ramp from black hole normal modes, JHEP 01 (2023) 153 [2208.14744]

    arXiv 2023

  4. [4]

    Das, S.K

    S. Das, S.K. Garg, C. Krishnan and A. Kundu,Fuzzballs and random matrices, JHEP 10 (2023) 031 [2301.11780]

    arXiv 2023

  5. [5]

    Das, S.K

    S. Das, S.K. Garg, C. Krishnan and A. Kundu,What is the Simplest Linear Ramp?, JHEP 01 (2024) 172 [2308.11704]

    arXiv 2024

  6. [6]

    Das and A

    S. Das and A. Kundu,Brickwall in rotating BTZ: a dip-ramp-plateau story, JHEP 02 (2024) 049 [2310.06438]

    arXiv 2024

  7. [7]

    Krishnan and P.S

    C. Krishnan and P.S. Pathak,Normal modes of the stretched horizon: a bulk mechanism for black hole microstate level spacing, JHEP 03 (2024) 162 [2312.14109]

    arXiv 2024

  8. [8]

    Burman, S

    V. Burman, S. Das and C. Krishnan,A smooth horizon without a smooth horizon, JHEP 2024 (2024) 014 [2312.14108]

    arXiv 2024

Show all 36 references
  1. [9]

    Banerjee, S

    S. Banerjee, S. Das, M. Dorband and A. Kundu,Brickwall, normal modes, and emerging thermality, Phys. Rev. D109 (2024) 126020 [2401.01417]

    2024 arXiv

  2. [10]

    Krishnan and R

    C. Krishnan and R. Mondol,Young Black Holes Have Smooth Horizons: A Swampland Argument, 2407.11952

    arXiv

  3. [11]

    Das,Stretched horizon from conformal field theory, JHEP 11 (2024) 033 [2406.10879]

    S. Das,Stretched horizon from conformal field theory, JHEP 11 (2024) 033 [2406.10879]. – 36 –

    2024 arXiv

  4. [12]

    Burman and C

    V. Burman and C. Krishnan,A Bottom-Up Approach to Black Hole Microstates, 2409.05850

    arXiv

  5. [13]

    S. Das, S. Porey and B. Roy,Brick Wall in AdS-Schwarzschild Black Hole: Normal Modes and Emerging Thermality, 2409.05519

    arXiv

  6. [14]

    Banerjee, S

    S. Banerjee, S. Das, A. Kundu and M. Sittinger,Blackish Holes, 2411.09500

    arXiv

  7. [15]

    Krishnan and P.S

    C. Krishnan and P.S. Pathak,Holomorphic Factorization at the Quantum Horizon, 2410.00732

    arXiv

  8. [16]

    Cotler, G

    J.S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S.H. Shenker et al.,Black Holes and Random Matrices, JHEP 05 (2017) 118 [1611.04650]

    2017 arXiv

  9. [17]

    Chen, H.W

    Y. Chen, H.W. Lin and S.H. Shenker,BPS chaos, SciPost Phys. 18 (2025) 072 [2407.19387]

    2025 arXiv

  10. [18]

    Mehta,On the statistical properties of the level-spacings in nuclear spectra, Nuclear Physics 18 (1960) 395

    M.L. Mehta,On the statistical properties of the level-spacings in nuclear spectra, Nuclear Physics 18 (1960) 395

    1960

  11. [19]

    Dyson,Statistical theory of the energy levels of complex systems

    F.J. Dyson,Statistical theory of the energy levels of complex systems. iii, Journal of Mathematical Physics 3 (1962) 166

    1962

  12. [20]

    Julia,STATISTICAL THEORY OF NUMBERS, inLes Houches School of Theoretical Physics: Number Theory and Physics, 9, 1989

    B. Julia,STATISTICAL THEORY OF NUMBERS, inLes Houches School of Theoretical Physics: Number Theory and Physics, 9, 1989

    1989

  13. [21]

    Bakas and M.J

    I. Bakas and M.J. Bowick,Curiosities of arithmetic gases, J. Math. Phys.32 (1991) 1881

    1991

  14. [22]

    Witten,Anti-de sitter space, thermal phase transition, and confinement in gauge theories, 1998

    E. Witten,Anti-de sitter space, thermal phase transition, and confinement in gauge theories, 1998

    1998

  15. [23]

    Aharony, J

    O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas and M. Van Raamsdonk,The Hagedorn - deconfinement phase transition in weakly coupled large N gauge theories, Adv. Theor. Math. Phys.8 (2004) 603 [hep-th/0310285]

    2004 arXiv

  16. [24]

    Sundborg,The Hagedorn transition, deconfinement and N=4 SYM theory, Nucl

    B. Sundborg,The Hagedorn transition, deconfinement and N=4 SYM theory, Nucl. Phys. B 573 (2000) 349 [hep-th/9908001]

    2000 arXiv

  17. [25]

    Titchmarsh,The Theory of the Riemann Zeta-Function, Oxford University Press, second edition, revised by d

    E.C. Titchmarsh,The Theory of the Riemann Zeta-Function, Oxford University Press, second edition, revised by d. r. heath-brown ed. (1986)

    1986

  18. [26]

    Gharibyan, M

    H. Gharibyan, M. Hanada, S.H. Shenker and M. Tezuka,Onset of Random Matrix Behavior in Scrambling Systems, JHEP 07 (2018) 124 [1803.08050]

    2018 arXiv

  19. [27]

    Nosaka, D

    T. Nosaka, D. Rosa and J. Yoon,The Thouless time for mass-deformed SYK, JHEP 09 (2018) 041 [1804.09934]

    2018 arXiv

  20. [28]

    What is an l-function?

    M.R. Murty, “What is an l-function?.” https://www.youtube.com/watch?v=922Jsbg487A, September, 2024. – 37 –

    2024

  21. [29]

    Peakmath landscape

    A. Holmström and the PeakMath Team, “Peakmath landscape.” https://www.youtube.com/@PeakMathLandscape, 2023

    2023

  22. [30]

    Iwaniec and E

    H. Iwaniec and E. Kowalski,Analytic Number Theory, vol. 53 ofAMS Colloquium Publications, American Mathematical Society (2004)

    2004

  23. [31]

    Montgomery and R.C

    H.L. Montgomery and R.C. Vaughan,Multiplicative Number Theory I: Classical Theory, Cambridge University Press (2007)

    2007

  24. [32]

    The L-functions and modular forms database

    The LMFDB Collaboration, “The L-functions and modular forms database.” https://www.lmfdb.org, 2025

    2025

  25. [33]

    Gubser, J

    S.S. Gubser, J. Knaute, S. Parikh, A. Samberg and P. Witaszczyk,p-adic ads/cft, Communications in Mathematical Physics352 (2017) 1019–1059

    2017

  26. [34]

    Maldacena, S.H

    J. Maldacena, S.H. Shenker and D. Stanford,A bound on chaos, Journal of High Energy Physics 2016 (2016)

    2016

  27. [35]

    Cogdell, H

    J. Cogdell, H. Kim and M. Murty,Lectures on Automorphic L-functions, Fields Institute monographs, American Mathematical Society (2004)

    2004

  28. [36]

    Gamma function properties: Asymptotic expansions

    NIST Digital Library of Mathematical Functions, “Gamma function properties: Asymptotic expansions.”https://dlmf.nist.gov/5.11, 2025. – 38 –

    2025

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.