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Positive quasimodular forms and the sign uncertainty principle

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Pith's one-line read For every dimension $d$ divisible by 4, the sign uncertainty constant satisfies $A_+(d) \le \sqrt{2\lfloor d/16\rfloor+2}$, recovering the optimal value at $d=12$ and improving the previous best bound for all $d \ge 52$.

desk verdict A new uniform upper bound for the sign uncertainty constant, with real machine-checked support for the key positivity lemmas, but the d ≡ 4 mod 8 half has omitted proofs that a referee should ask to see. read the letter →

arxiv 2608.15415 v2 pith:5FZBADGN submitted 2026-08-15 math.NT math.CA

Seewoo Lee This is my paper · ORCID
classification math.NTmath.CA MSC 11F0311F1111F3042A38
keywords signuncertaintyprincipleBourgain–Clozel–KahaneconstantquasimodularformsextremalFouriereigenfunctionsmodularlineardifferentialequationshypergeometricseriespositivityontheimaginaryaxis
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 proves a new upper bound on the Bourgain–Clozel–Kahane sign uncertainty constant $A_+(d)$: for every positive integer $d$ divisible by 4, $A_+(d) \le \sqrt{2\lfloor d/16\rfloor+2}$. This improves the previously best explicit uniform bound $\sqrt{(d+2)/(2\pi)}$ for all $d \ge 52$ divisible by 4 and recovers the known optimal value $A_+(12)=\sqrt{2}$. The proof constructs explicit self-Fourier functions whose last sign change occurs at the claimed radius, and the whole argument reduces to showing certain quasimodular forms are positive on the imaginary axis. The argument splits into two cases according to $d \bmod 8$, each reducing positivity of a natural family of quasimodular forms to positivity of the depth-2 extremal quasimodular forms of the Kaneko–Koike type.

What carries the argument

The central mechanism is a Laplace-transform bridge from positive quasimodular forms to Fourier eigenfunctions. For the $+1$-eigenfunction family, the integral $M_{d,+}(x)=4\sin^2(\pi\|x\|^2/2)\int_0^\infty t^{2-w} F_w(i/t)\,\Delta(it)^{-n_{d,+}} e^{-\pi\|x\|^2 t}\,dt$ turns a positive weight-$w$ depth-2 form $F_w$ into an eigenfunction that is eventually nonnegative and vanishes at $\|x\|=\sqrt{2n}$ for every integer $n>n_{d,+}$. The sign of $M_{d,+}(0)$ is controlled by the leading coefficient of $\delta F_w = \partial F_w/\partial E_2$, a derived depth-1 cusp form; the paper proves those coefficients are positive. Positivity of $F_w$ is reduced by the identities $F_w = -C_w X_{w,2}+E_4 X_{w-4,2}$ and $F_{w+2}=A_w E_4 X_{w-2,2}+B_w E_6 X_{w-4,2}$ to the positivity of the depth-2 extremal forms $X_{w,2}$, which is established through the recurrence $X_{w+2,2}\propto E_4 X_{w-2,2}-E_6 X_{w-4,2}$ together with hypergeometric series expansions for $X_{w,2}$. The $d\equiv4\pmod8$ half uses a level-2 companion family $Y_w$ satisfying the identical shifted relations, positivity of which is proved by hypergeometric expansions of the same $\,_3F_2$ type plus elementary log inequalities; a separate coefficient-positivity induction for the derived form $\widetilde G_w$ (Propositions 4.23–4.24) supplies the strict sign of $M_{d,-}(0)$.

What would settle it

Evaluate $Y_w(it)$ numerically for a weight beyond the base cases, say $w=14$, at $t=1.05$ using the recurrence (81)–(82); a negative value would falsify Theorem 4.18. For the $d\equiv0\pmod8$ half, compute the coefficient $b_{0,+}$ given by (58)–(60) at $d=16$; positivity of that coefficient is necessary for Corollary 4.10, and the formula is explicit enough for a direct finite computation.

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Extended reading notes

Core claim

Theorem 1.1 states that for every positive integer $d$ divisible by 4, $A_+(d) \le \sqrt{2\lfloor d/16\rfloor+2}$, with strict inequality unless $d=12$, where $A_+(12)=\sqrt{2}$. When $d \equiv 0 \pmod 8$, the $+1$-eigenfunction $M_{d,+}$ is built from the depth-2 quasimodular forms $F_w$ of the Feigenbaum–Grabner–Hardin family, and the paper proves each $F_w$ is positive on the positive imaginary axis by relating it to the Kaneko–Koike depth-2 extremal forms $X_{w,2}$ through new identities; positivity of $X_{w,2}$ then follows from new recurrences and Nakaya's hypergeometric identity. When $d \equiv 4 \pmod 8$, the $(-1)$-eigenfunction $M_{d,-}$ is built from a level-$\Gamma(2)$ companion family $G_w$, and the paper introduces forms $Y_w$ satisfying the same shifted differential and recurrence relations as the $X_{w,2}$; positivity of $Y_w$ and $G_w$ on the imaginary axis is proved using hypergeometric expansions of the same type together with elementary log inequalities. In both halves, coefficient positivity of the derived forms $\delta F_w$ and $\widetilde G_w$ forces $M(0)<0$, so that $f = M - M(0)e^{-\pi\|x\|^2}$ lies in $\mathcal A_+(d)$ with last-sign-change radius $\sqrt{2\lfloor d/16\rfloor+2}$.

Load-bearing premise

The load-bearing premise is that the companion forms $Y_w$—and hence the level-2 forms $G_w$ they feed—are positive on the imaginary axis, a positivity whose key hypergeometric expansions are asserted with only a 'similar' proof and whose auxiliary identities are summarized as routine.

Editorial extensions

If this is right

  • For all $d \ge 52$ divisible by 4, the new explicit bound $A_+(d) \le \sqrt{2\lfloor d/16\rfloor+2}$ supersedes the previous best uniform bound $\sqrt{(d+2)/(2\pi)}$.
  • The known exact optimum at $d=12$, $A_+(12)=\sqrt{2}$, is recovered, and for $d\equiv4\pmod8$ except $d=12$ the inequality is strict, certifying that $A_+(d)$ lies strictly below the stated square root.
  • The construction provides a uniform family of explicit self-Fourier functions with last-sign-change radius $\sqrt{2\lfloor d/16\rfloor+2}$, giving concrete test functions for numerical study of the sign uncertainty constant in arbitrarily high dimensions.
  • As a byproduct, the paper proves a weak form of the Kaneko–Koike positivity conjecture for depth 2: the extremal quasimodular forms $X_{w,2}$ are positive on the imaginary axis for every even weight $w\ge4$, $w\neq6$.
  • The bound has asymptotic order $\sqrt{2\lfloor d/16\rfloor+2}=(1/\sqrt8+o(1))\sqrt d\approx0.354\sqrt d$, within 11\% of the recently established limit $1/\pi\approx0.318$ for $A_+(d)/\sqrt d$.
  • The paper's positivity results suggest the stronger complete-positivity statements conjectured in Remarks 4.11, 4.20, and 4.26; proving them would remove the analytic-continuation step from the construction and likely yield a fully coefficient-level proof.
  • The same machinery of positive quasimodular forms producing Fourier eigenfunctions should give explicit upper bounds on the companion constant $A_-(d)$ (defined by $(-1)$-eigenfunctions), since the construction only requires the sign of $M(0)$ relative to the eventual nonnegativity of $M$.
  • The dependence on $\lfloor d/16\rfloor$ suggests the bound might be improvable by optimizing the parameters $n_{d,\pm}$ or by using deeper extremal quasimodular forms (depth $>2$), a direction the paper leaves open.
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Editorial analysis

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Referee Report

3 major / 3 minor

Summary. The paper proves a new explicit upper bound A_+(d) ≤ sqrt(2 floor(d/16) + 2) for the Bourgain–Clozel–Kahane sign uncertainty constant for every positive integer d divisible by 4. The bound recovers the optimal value A_+(12) = sqrt(2) and improves the previously best known explicit bound sqrt((d+2)/(2π)) for all d ≥ 52 divisible by 4. The proof constructs Feigenbaum–Grabner–Hardin Fourier eigenfunctions whose last-sign-change radius is sqrt(2 floor(d/16) + 2), and reduces the required positivity of the associated quasimodular forms to positivity of Kaneko–Koike extremal forms and a new level-2 companion family Y_w. The d ≡ 0 (mod 8) case is supported by coefficient-positivity proofs, partly formalized in Lean, while the d ≡ 4 (mod 8) case relies on the companion family Y_w and a hypergeometric expansion that is only sketched.

Significance. If the proof is completed, the result is a substantial improvement: it gives an explicit uniform upper bound of order sqrt(d) with a smaller constant than the previous explicit bound, and it places the optimal d = 12 construction within a general framework. The paper is commendably transparent about the computational assistance it uses, and it ships machine-checked Lean proofs for Lemma 2.4, Proposition 4.9, and Propositions 4.23–4.24, together with Sage verifications of the family identities up to weight 100. The main theorem is parameter-free and the target bound is not assumed, so there is no circularity in the argument. The central concern is the companion family Y_w: Proposition 4.16 and parts of Proposition 4.15 are load-bearing and under-proved, and the d ≡ 4 (mod 8) half of Theorem 1.1 depends on them.

major comments (3)
  1. [§4.2, Proposition 4.16] Proposition 4.16 asserts the hypergeometric expansions (92) and (93) with the proof "similar to that of Theorem 3.4". This is not a routine analogue: by Proposition 4.15(4), Y_w = eY_w L_S + Φ_w, so Y_w is not modular and h = E4^{-w/4} Y_w is not a function of x = 1728/j on the upper half-plane. The reduction of the MLDE (83) to the hypergeometric ODE (91) in the variable x, the uniqueness of the local solution with the stated exponent, and the branch/continuation behavior at t = 1 for the divergent series in (93) are exactly the steps that need a detailed proof. These formulas are the sole justification for positivity of Y_w for t > 1 in Theorem 4.18, which in turn feeds Corollary 4.19, Corollary 4.25, and the d ≡ 4 (mod 8) half of Theorem 1.1.
  2. [§4.2, Propositions 4.13 and 4.15(2)–(4)] Propositions 4.13 and 4.15(2)–(4) state the identities (78)–(79) and (85)–(90) with the comments "we omit the details of the proof" and "we omit the routine details". These identities are used in the induction in Theorem 4.18 (via (88) and (82)) and in Corollary 4.19 (via (89)–(90) and (79)). Because the logarithmic term L_S appears in G_w and Y_w, these recurrences are not routine translations of the quasimodular case; the manuscript needs to provide the derivations or a rigorous computational verification that covers the full weight range and justifies the differentiation of the L_S terms.
  3. [§4.2, Corollary 4.19] Corollary 4.19 asserts "we can still apply Proposition 3.2(4) even if G_w are not quasimodular forms in general" without proof. Proposition 3.2(4) is stated for quasimodular forms, and the extension to functions of the form eG_w Δ L_S + Ψ_w requires an argument, for example the monotonicity of G_w/Δ^{w/12} analogous to the one given for Y_w in the proof of Theorem 4.18. This extension is needed to conclude positivity of G_w for 0 < t < 1 from positivity of G_{w+2}, which is essential for the eventual nonnegativity of M_{d,-} in Corollary 4.25.
minor comments (3)
  1. [§4.2, Proof of Theorem 4.18] In the sentence "If Y_{w+2} is positive, then ∂_w Y_w is also positive by (88)", the reference appears to be to (82) rather than (88); please correct the reference.
  2. [§4.1.2] The discussion of the analytic continuation of (44) would benefit from stating explicitly why the subtracted terms in (57) have exponential decay and why the remaining integral converges near t = 0; currently this is only implicit.
  3. [§4.2, Theorem 4.12] The range for identity (73) excludes w = 16 because of the factor (w−16) in the denominator, but this is not mentioned where the recurrence is later used in Proposition 4.13; a brief remark would avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the target bound is never assumed, and the proof reduces to independent positivity inputs and explicit finite data rather than to its own conclusion.

full rationale

I found no step in which a claimed prediction is equivalent by construction to an input, or in which the target inequality A_+(d) <= sqrt(2 floor(d/16)+2) is assumed. The square-root constants are not fitted: they come from the FGH Fourier eigenfunction construction, where the vanishing order of F_w (resp. G_w) at the cusp and the zeros of sin^2(pi ||x||^2/2) determine the last-sign-change radius sqrt(2 n_{d,+}) (resp. sqrt(2 n_{d,-})), with n_{d,+}, n_{d,-} defined directly from w and d. Positivity of F_w is reduced, via explicit identities such as (47), (48), (51), (52), to positivity of the depth-2 extremal quasimodular forms X_{w,2}; that positivity is proved from Nakaya's independent hypergeometric identity, Pellarin's uniqueness theorem, Kaneko-Koike recurrences, and finite low-weight base cases, not from the bound being proved. Positivity of G_w is similarly reduced to the companion family Y_w; the main weakness there is that Proposition 4.16 is asserted with 'the proof is similar to that of Theorem 3.4' and Proposition 4.15 omits routine induction details. That is a correctness gap or a missing proof, not a circular reduction: the hypergeometric expansions are not obtained by assuming G_w positivity or by assuming Theorem 1.1. The self-citations to the author's earlier paper [16] supply a general positivity-transfer lemma (Proposition 3.2) and finite base cases; these do not encode the target constant and are externally verifiable. The result A_+(12)=sqrt(2) is quoted only as context and the proof of Theorem 1.1 does not use its lower bound. The Lean formalizations of Lemmas 2.2-2.4 and of the coefficient positivity of eF and eG, together with Sage checks, are independent support. Overall, the derivation chain is self-contained; the d ≡ 4 mod 8 branch contains omitted justifications, but no step reduces to its own output by definition.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

There are no free parameters fitted to data, and no new postulated mathematical objects such as new constants, dimensions, or operators. The proof rests on standard quasimodular-form theory, the external eigenfunction construction of Feigenbaum-Grabner-Hardin, hypergeometric identities of Nakaya, and base cases from the author's earlier work.

assumptions (5)
  • standard math Ramanujan's differential identities (8) for E2, E4, E6
    Invoked throughout Section 2 and taken as axioms in the Lean formalization (Appendix A.1). These are standard theorems of the theory of quasimodular forms, not derived in this paper.
  • standard math Algebraic independence of E2, E4, E6 over C and the polynomial representation of the quasimodular ring
    Used in Section 2.1 and in the Lean polynomial model (Appendix A), cited to Martin-Royer [17, Lemma 117].
  • domain assumption Feigenbaum-Grabner-Hardin theorem: the integral transforms M_{d,+} and M_{d,-} are Fourier eigenfunctions with the stated zero sets and analytic continuation
    External theorem from [7], restated as Theorems 4.1 and 4.12. The new proof of Theorem 1.1 builds on this construction rather than proving it from scratch.
  • domain assumption Nakaya's hypergeometric formulas for normalized extremal quasimodular forms of depth 2
    Used to prove positivity of X_{w,2} (Theorem 3.4, with a proof supplied) and, by analogy, of Y_w (Proposition 4.16, proof only sketched). This underpins the t>1 positivity arguments.
  • domain assumption Base-case complete positivity and positivity transfer facts (Proposition 3.2) from Lee [16]
    The paper cites the author's earlier work for admissible low-weight complete positivity and for derivative/positivity results used in Theorems 3.5, 4.5, and 4.18. This is a self-citation but not a circular use of the target result.

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Pith. "Pith review of Positive quasimodular forms and the sign uncertainty principle." pith.science (2026). https://pith.science/paper/5FZBADGN

@misc{pith2026260815415,
  author       = {Pith},
  title        = {Pith review of: Positive quasimodular forms and the sign uncertainty principle},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5FZBADGN}},
  note         = {Machine review of arXiv:2608.15415}
}
abstract

For every positive integer $d$ divisible by $4$, we prove the following new upper bound for the Bourgain-Clozel-Kahane sign uncertainty constant: \[ \mathrm{A}_+(d) \le \sqrt{2 \left\lfloor \frac{d}{16} \right\rfloor + 2}. \] It recovers the optimal bound $\mathrm{A}_+(12) \le \sqrt{2}$ in dimension $12$ and improves the previously best known bound $\sqrt{(d+2)/(2\pi)}$ for all $d \ge 52$ divisible by $4$. The proof uses Fourier eigenfunctions and associated quasimodular forms constructed by Feigenbaum, Grabner, and Hardin.

Figures

Figures reproduced from arXiv: 2608.15415 by the authors.

Figure 1
Figure 1. Upper bounds on A+(𝑑), with a zoomed view for 𝑑 ≤ 100 in the inset. The teal dots are the numerical polynomial–Gaussian bounds of Cohn–Gonçalves for 1 ≤ 𝑑 ≤ 32 [5, [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

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Works this paper leans on

26 extracted references · 23 canonical work pages

  1. [1]

    NimaAfkhami-Jeddi,HenryCohn,ThomasHartman,DaviddeLaat,andAmirhosseinTajdini.High-dimensionalspherepacking and the modular bootstrap.Journal of High Energy Physics, 2020(12):1–45, 2020

    work page 2020

  2. [2]

    Principe d’Heisenberg et fonctions positives.Annales de l’Institut Fourier, 60(4):1215–1232, 2010

    Jean Bourgain, Laurent Clozel, and Jean-Pierre Kahane. Principe d’Heisenberg et fonctions positives.Annales de l’Institut Fourier, 60(4):1215–1232, 2010

    work page 2010

  3. [3]

    Elliptic modular forms and their applications.The 1-2-3 of modular forms: Lectures at a summer school in Nordfjordeid, Norway, pages 1–103, 2008

    Jan Hendrik Bruinier, Gerard van der Geer, Günter Harder, and Don Zagier. Elliptic modular forms and their applications.The 1-2-3 of modular forms: Lectures at a summer school in Nordfjordeid, Norway, pages 1–103, 2008

    work page 2008

  4. [4]

    Sign uncertainty principles and low-degree polynomials.Proceedings of the American Mathematical Society, Series B, 11(21):224–228, 2024

    Henry Cohn, Dingding Dong, and Felipe Gonçalves. Sign uncertainty principles and low-degree polynomials.Proceedings of the American Mathematical Society, Series B, 11(21):224–228, 2024

    work page 2024

  5. [5]

    An optimal uncertainty principle in twelve dimensions via modular forms.Inventiones mathematicae, 217:799–831, 2019

    Henry Cohn and Felipe Gonçalves. An optimal uncertainty principle in twelve dimensions via modular forms.Inventiones mathematicae, 217:799–831, 2019

    work page 2019

  6. [6]

    The sphere packing problem in dimension 24.Annals of Mathematics, 185(3):1017–1033, 2017

    Henry Cohn, Abhinav Kumar, Stephen Miller, Danylo Radchenko, and Maryna Viazovska. The sphere packing problem in dimension 24.Annals of Mathematics, 185(3):1017–1033, 2017

    work page 2017

  7. [7]

    Eigenfunctions of the Fourier Transform with specified zeros

    Ahram S Feigenbaum, Peter J Grabner, and Douglas P Hardin. Eigenfunctions of the Fourier Transform with specified zeros. Mathematical Proceedings of the Cambridge Philosophical Society, 171(2):329–367, 2021

    work page 2021

  8. [8]

    Hermite polynomials, linear flows on the torus, and an uncertainty principle for roots.Journal of Mathematical Analysis and Applications, 451(2):678–711, 2017

    Felipe Gonçalves, Diogo Oliveira e Silva, and Stefan Steinerberger. Hermite polynomials, linear flows on the torus, and an uncertainty principle for roots.Journal of Mathematical Analysis and Applications, 451(2):678–711, 2017

    work page 2017

Show all 26 references
  1. [9]

    Quasimodular forms as solutions of modular differential equations.International Journal of Number Theory, 16(10):2233–2274, 2020

    Peter J Grabner. Quasimodular forms as solutions of modular differential equations.International Journal of Number Theory, 16(10):2233–2274, 2020

    2020

  2. [10]

    The Ramanujan Journal, 57(3):1021–1041, 2022

    PeterJGrabner.AsymptoticexpansionsforthecoefficientsofextremalquasimodularformsandaconjectureofKanekoandKoike. The Ramanujan Journal, 57(3):1021–1041, 2022

    2022

  3. [11]

    Coefficient Bounds for Level 2 Cusp Forms and Modular Functions.arXiv preprint arXiv:1408.1083, 2014

    Paul Jenkins and Kyle Pratt. Coefficient Bounds for Level 2 Cusp Forms and Modular Functions.arXiv preprint arXiv:1408.1083, 2014

    2014 arXiv

  4. [12]

    Bounds for coefficients of cusp forms and extremal lattices.Bulletin of the London Mathematical Society, 43(5):927–938, 2011

    Paul Jenkins and Jeremy Rouse. Bounds for coefficients of cusp forms and extremal lattices.Bulletin of the London Mathematical Society, 43(5):927–938, 2011

    2011

  5. [13]

    On extremal quasimodular forms.Kyushu Journal of Mathematics, 60(2):457–470, 2006

    Masanobu Kaneko and Masao Koike. On extremal quasimodular forms.Kyushu Journal of Mathematics, 60(2):457–470, 2006

    2006

  6. [14]

    MasanobuKaneko,KiyokazuNagatomo,andYuichiSakai.Thethirdordermodularlineardifferentialequations.JournalofAlgebra, 485:332–352, 2017

    2017

  7. [15]

    Supersingular j-invariants, hypergeometric series, and Atkin’s orthogonal polynomials

    Masanobu Kaneko and Don Zagier. Supersingular j-invariants, hypergeometric series, and Atkin’s orthogonal polynomials. AMS/IP Studies in Advanced Mathematics, 7:97–126, 1998

    1998

  8. [16]

    Algebraic proof of modular form inequalities for optimal sphere packings.arXiv preprint arXiv:2406.14659, 2024

    Seewoo Lee. Algebraic proof of modular form inequalities for optimal sphere packings.arXiv preprint arXiv:2406.14659, 2024

    2024 arXiv

  9. [17]

    Société Mathématique de France, Paris, 2005

    FrançoisMartinandEmmanuelRoyer.Formesmodulairesetpériodes.InFormesmodulairesettranscendance,volume12ofSéminaires et Congrès, pages 1–117. Société Mathématique de France, Paris, 2005

    2005

  10. [18]

    The Lean 4 theorem prover and programming language

    Leonardo de Moura and Sebastian Ullrich. The Lean 4 theorem prover and programming language. InInternational Conference on Automated Deduction, pages 625–635. Springer, 2021

    2021

  11. [19]

    Transactions of the American Mathematical Society, 377(11):7657–7698, 2024

    KiyokazuNagatomo,YuichiSakai,andDonZagier.ModularlineardifferentialoperatorsandgeneralizedRankin-Cohenbrackets. Transactions of the American Mathematical Society, 377(11):7657–7698, 2024

    2024

  12. [20]

    Tomoaki Nakaya. Determination of normalized extremal quasimodular forms of depth 1 with integral fourier coefficients.Inter- national Journal of Number Theory, 20(03):641–689, 2024.arXiv:2305.18669

    2024 arXiv

  13. [21]

    Alexander Novikov, Ngân V˜u, Marvin Eisenberger, Emilien Dupont, Po-Sen Huang, Adam Zsolt Wagner, Sergey Shirobokov, Borislav Kozlovskii, Francisco J. R. Ruiz, Abbas Mehrabian, M. Pawan Kumar, Abigail See, Swarat Chaudhuri, George Holland, Alex Davies, Sebastian Nowozin, Pushm...

    2025

  14. [22]

    Ten Advances in Mathematics and Theoretical Computer Science.https://cdn.openai.com/pdf/ten-proofs-oai.pdf

    OpenAI. Ten Advances in Mathematics and Theoretical Computer Science.https://cdn.openai.com/pdf/ten-proofs-oai.pdf. Accessed: 2026-08-09

    2026

  15. [23]

    On extremal quasi-modular forms after Kaneko and Koike.Kyushu Journal of Mathematics, 74(2):401–413, 2020

    Federico Pellarin. On extremal quasi-modular forms after Kaneko and Koike.Kyushu Journal of Mathematics, 74(2):401–413, 2020. With an appendix by Gabriele Nebe

    2020

  16. [24]

    Modular forms of half integral weight

    Goro Shimura. Modular forms of half integral weight. InModular Functions of One Variable I: Proceedings International Summer School University of Antwerp, RUCA July 17–August 3, 1972, pages 57–74. Springer, 1973

    1972

  17. [25]

    The Sage Developers.SageMath, the Sage Mathematics Software System (Version 10.9), 2026.https://www.sagemath.org

    2026

  18. [26]

    The sphere packing problem in dimension 8.Annals of mathematics, pages 991–1015, 2017

    Maryna Viazovska. The sphere packing problem in dimension 8.Annals of mathematics, pages 991–1015, 2017. 36

    2017

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