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arxiv: 2605.09731 · v1 · submitted 2026-05-10 · 🧮 math.NT

Recognition: 2 theorem links

· Lean Theorem

The variation of zeros of the Miller basis

Liubomir Chiriac , Andrei Jorza
Authors on Pith no claims yet

Pith reviewed 2026-05-12 03:59 UTC · model grok-4.3

classification 🧮 math.NT
keywords modular formsMiller basiszerosSzegő curveq-expansionsunit circlealgebraic zerosnumber theory
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The pith

Zeros of Miller basis modular forms lie on the unit arc or a log Szegő curve depending on the ratio δ = m/ℓ.

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

The paper links the zeros of modular forms in the Miller basis, whose q-expansions begin with q^m plus higher-order terms up to q^{ℓ+1}, to a logarithmic variant of the Szegő curve that depends on the ratio δ = m/ℓ. For δ below roughly 0.6194 the zeros all sit on the unit arc once the weight k grows large enough, while for δ near 1 they lie instead on the curve S_δ. The authors conjecture that the zeros always occupy the union of the unit arc and S_δ for any δ and prove a partial case of this statement. They also compute all algebraic zeros that appear for small values of ℓ minus m.

Core claim

We exhibit a connection between the variation of zeros in the Miller basis of modular forms q^m + O(q^{ℓ+1}) and a logarithmic version S_δ of the Szegő curve, where δ = m/ℓ. When δ < 0.6194 we show that all the zeros are on the unit arc for k ≫ 0, while if δ is asymptotically close to 1, we show that all the zeros lie on S_δ. In general, we posit that for all δ, the zeros are located on the union of the unit arc and the log Szegő curve, obtaining a partial result, and find conjectural thresholds for m/ℓ with all zeros on the unit arc, and no zeros on the arc. Finally, we enumerate all algebraic zeros of Miller forms up to ℓ-m ≤ 25.

What carries the argument

The logarithmic Szegő curve S_δ defined by the ratio δ = m/ℓ, which determines the limiting locus of the zeros of the Miller basis forms.

If this is right

  • When δ is less than about 0.6194, all zeros of the Miller forms remain on the unit arc for all sufficiently large weights.
  • When δ is asymptotically close to 1, the zeros of the Miller forms lie entirely on the logarithmic Szegő curve S_δ.
  • There exist conjectural threshold values of δ that mark the change from all zeros on the unit arc to none on the arc.
  • All algebraic zeros among Miller forms with ℓ − m at most 25 can be listed explicitly.
  • A partial proof supports the claim that zeros always lie on the union of the unit arc and S_δ.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same ratio-based description may apply to zero loci in other explicit bases of modular forms.
  • The complete list of algebraic zeros for small cases could be used to study associated Galois representations or L-values.
  • Systematic numerical checks at intermediate values of δ could tighten the conjectural thresholds.
  • The method might extend to q-expansions with more than two leading terms or to forms on higher-genus curves.

Load-bearing premise

The positions of the zeros are governed only by the ratio δ together with the large-weight asymptotic behavior, without extra terms in the q-expansion moving them elsewhere.

What would settle it

Numerically locate all zeros of a Miller basis form with δ = 0.5 and weight k = 200, then check whether every zero lies exactly on the unit circle.

Figures

Figures reproduced from arXiv: 2605.09731 by Andrei Jorza, Liubomir Chiriac.

Figure 1
Figure 1. Figure 1: The Szeg˝o curve The logarithmic Szeg˝o curve L± = {τ ∈ H : | Re τ | ≤ 1/2, ±24e 2πiτ ∈ S} is the graph of the function g± : [−0.5, 0.5] → H given by g±(x) = ln 24 − ln u −1 (± cos(2πx)) 2π , where u : [W(e −1 ), 1] → [−1, 1] is the bijection u(x) = (1 + ln x)/x. 2 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Asymptotic location of zeros as δ → 1 The Szeg˝o curve arises naturally as the limiting locus of the zeros of truncated exponential polynomials under an appropriate normalization. A noteworthy aspect of our theorem is that the relevant truncation threshold is governed solely by the ratio m/ℓ, as in the preceding result. This observation leads us to formulate the following general conjecture. Conjecture C. … view at source ↗
Figure 3
Figure 3. Figure 3: Contour of integration This yields (2.1) gk,m(e iθ) = Z 1 2 +iB − 1 2 +iB G(τ, eiθ)dτ − 2πiX γ Resτ=γeiθ G(τ, eiθ), 4 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Choice of Br for each arc Ar Suppose, now, that δ < 3 π ≈ 0.9549. Since δr is decreasing in r, there exists a maximum r0 such that δ < δr for all r ≤ r0. We conclude that gk,m(e iθ) = 2 cos(kθ/2 + 2πm cos θ) + o(1) for all θ ∈ [ π 2 , βr0 ]. Then Proposition 4 applies for this interval and so the Miller form gk,m will have at least  kβr0 2π + 2m cos βr0  −  k 4  roots on the arc A, all of them lying on… view at source ↗
Figure 5
Figure 5. Figure 5: Lower bound P for the proportion of roots on A The first lower bound comes from the fact that the linear function connecting (0.6194, 1) and (0.9546, 0) lies below the graph of P. We may similarly define a function T (δ) by setting T (δ) = 2π 3 for δ < 0.6194, T (δ) = βr0 if δr0+1 ≤ δ < δr0 , 3 π , and T (δ) = π 2 otherwise. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Angle T (δ) with Aπ 2 ,T (δ) containing Miller zeros By the above, gk,m(e iθ) = 2 cos(kθ/2 + 2πm cos θ) + o(1) on the arc [ π 2 , T (δ)] and the equidistribution was proved in [Rav25, §3.4] on any arc where this asymptotics holds. The conclusion follows form the fact that Θ is the piece-wise linear lower hull of T . □ 4. Zeros of weakly holomorphic forms In this section, we extend our previous result to we… view at source ↗
Figure 7
Figure 7. Figure 7: Lower bound P− for the proportion of roots on A 12 [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Angle T−(δ) with Aπ 2 ,T−(δ) containing Miller zeros 5. Proof of Theorem B Consider again the Miller basis gk,m = q m + O(q ℓ+1) of weight k = 12ℓ + k ′ , with k ′ ∈ {0, 4, 6, 8, 10, 14}. For some monic polynomial P of degree D = ℓ − m we have gk,m ∆ℓEk ′ = P(j), where j = E 3 4 /∆ is the usual j-invariant. We refer to P as the Faber polynomial of gk,m. Writing P(X) = X D d=0 ydx D−d y0 = 1, we will be con… view at source ↗
Figure 9
Figure 9. Figure 9: Asymptotic location of zeros as δ → 1 Proof. Let ED(x) = P D k=0 akx D−k , where ak = 1 k! , in which case, by Proposition 11 applied to ε ∈ (α/2, 1/2), the Faber polynomial satisfies F(2kx) (2k)D = X D k=0 bkx D−k , where bk = ak(1+O(k −α )). As in [Rud24, §5.1], we use Ostrowski’s theorem which states that the roots of ED(x) and F(2kx) are within 2DΓ(P|ak−bk|Γ −k ) 1/D, where Γ = max(a 1/k k , b1/k k ). … view at source ↗
Figure 10
Figure 10. Figure 10: Roots and Sδ for several values of δ ∈ (0, 1) 19 [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Roots and Sδ for several values of δ ∈ (1,∞) Remark 2. Let δ + A = 1 − 24 W(e −1 )e √ 3π = 0.6265 . . . δ− A = 1 + 24 W(e −1 )e 2π = 1.1609 . . . δ + S = 1 − 24 e 2π = 0.9551 . . . δ− S = 1 + 24 e √ 3π = 1.1040 . . . The observation before Theorem B implies the following. (1) If δ ∈ (0, 1) then Cδ = A if δ < δ+ A and Cδ = Sδ if δ > δ+ S . If δ ∈ (δ + A, δ+ S ), then Cδ contains Aπ 2 ,Tδ , where − cos Tδ i… view at source ↗
Figure 12
Figure 12. Figure 12: Conjectural value of Tδ with Aπ 2 ,T−(δ) containing Miller zeros The conjectural value of Tδ is very close to the computational values of T (δ) in [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
read the original abstract

We exhibit a connection between the variation of zeros in the Miller basis of modular forms $q^m+O(q^{\ell+1})$ and a logarithmic version $\mathcal{S}_\delta$ of the Szeg\H{o} curve, where $\delta=m/\ell$. When $\delta<0.6194$ we show that all the zeros are on the unit arc for $k\gg 0$, while if $\delta$ is asymptotically close to 1, we show that all the zeros lie on $\mathcal{S}_{\delta}$. In general, we posit that for all $\delta$, the zeros are located on the union of the unit arc and the log Szeg\H{o} curve, obtaining a partial result, and find conjectural thresholds for $m/\ell$ with all zeros on the unit arc, and no zeros on the arc. Finally, we enumerate all algebraic zeros of Miller forms up to $\ell-m\leq 25$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies the zeros of the Miller basis elements in spaces of modular forms, which admit q-expansions of the form q^m + O(q^{ℓ+1}) with parameter δ = m/ℓ. It connects the zero loci to a logarithmic variant S_δ of the Szegő curve. Rigorous proofs are given that all zeros lie on the unit arc when δ < 0.6194 and k ≫ 0, and that all zeros lie on S_δ when δ is asymptotically close to 1. The authors conjecture that for general δ the zeros lie on the union of the unit arc and S_δ, obtain a partial result toward this, state conjectural thresholds separating regimes with all zeros on the arc versus none on the arc, and computationally enumerate all algebraic zeros for ℓ − m ≤ 25.

Significance. If the results hold, the work provides a new analytic link between zero distributions of a distinguished basis of modular forms and classical Szegő curves in logarithmic form. The rigorous proofs for the two extreme regimes of δ, together with the explicit computational enumeration of algebraic zeros, constitute concrete strengths; these are parameter-free in the sense that they derive directly from the definition of δ and the q-expansion shape without fitted constants.

major comments (2)
  1. [Section on asymptotic analysis for general δ (near the statement of the partial result)] The partial result toward the general conjecture (zeros on the union of the unit arc and S_δ) rests on large-k asymptotic analysis of the Miller basis expansions. This analysis lacks uniform error bounds on the O(q^{ℓ+1}) remainder that hold across the full range of intermediate δ; without such uniformity the extension from the extreme regimes to the posited union cannot be considered load-bearing for the central claim.
  2. [The paragraph defining S_δ and the subsequent theorem for δ asymptotically close to 1] The definition of the logarithmic Szegő curve S_δ is introduced directly from δ, but the justification that it accurately captures the zero loci for δ near 1 relies on coefficient growth estimates whose uniformity in δ is not established; this is load-bearing for the claim that all zeros lie on S_δ in that regime.
minor comments (2)
  1. [Abstract] The abstract uses both 'log Szegő curve' and 'logarithmic version S_δ of the Szegő curve'; consistent terminology throughout the text would improve readability.
  2. [The final enumeration section] The computational enumeration up to ℓ − m ≤ 25 is presented without an explicit statement of the precision or software used to locate the algebraic zeros; adding this would strengthen the supporting data.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for highlighting the strengths of the rigorous results in the extreme regimes of δ together with the computational enumeration. We address each major comment below with clarifications on the scope of our claims and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Section on asymptotic analysis for general δ (near the statement of the partial result)] The partial result toward the general conjecture (zeros on the union of the unit arc and S_δ) rests on large-k asymptotic analysis of the Miller basis expansions. This analysis lacks uniform error bounds on the O(q^{ℓ+1}) remainder that hold across the full range of intermediate δ; without such uniformity the extension from the extreme regimes to the posited union cannot be considered load-bearing for the central claim.

    Authors: The manuscript explicitly labels this as a partial result and does not claim that the asymptotic analysis establishes the full conjecture for all intermediate δ. The large-k analysis is used only to obtain supporting evidence in certain subranges where the remainder can be controlled for fixed δ. We agree that the lack of fully uniform bounds across all intermediate δ limits the strength of this evidence. In the revision we will add a clarifying paragraph stating the precise conditions under which the O(q^{ℓ+1}) terms are estimated and explicitly note that uniformity over the entire interval of δ is not claimed. This does not alter the rigorous theorems for the extreme regimes. revision: partial

  2. Referee: [The paragraph defining S_δ and the subsequent theorem for δ asymptotically close to 1] The definition of the logarithmic Szegő curve S_δ is introduced directly from δ, but the justification that it accurately captures the zero loci for δ near 1 relies on coefficient growth estimates whose uniformity in δ is not established; this is load-bearing for the claim that all zeros lie on S_δ in that regime.

    Authors: The theorem is stated only for δ asymptotically close to 1, and the coefficient growth estimates are obtained under this asymptotic assumption, which permits uniformity within a sufficiently small neighborhood of δ = 1. The definition of S_δ is obtained by direct logarithmic transformation of the classical Szegő curve and does not itself require additional uniformity. To make the justification fully explicit we will insert a short lemma verifying that the relevant growth bounds hold uniformly for δ in the stated asymptotic regime. This revision will be included in the next version. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent asymptotic analysis and explicit definitions.

full rationale

The paper defines the Miller basis explicitly as forms with q-expansion q^m + O(q^{ℓ+1}) and introduces the logarithmic Szegő curve S_δ directly from the ratio δ = m/ℓ. The main results consist of analytic proofs that zeros lie on the unit arc for δ < 0.6194 (large k) and on S_δ when δ is near 1, plus a partial result and conjecture for the union, supported by explicit enumeration of algebraic zeros up to ℓ-m ≤ 25. No step reduces a claimed prediction to a fitted input by construction, nor does any load-bearing premise collapse to a self-citation or self-definitional loop. The asymptotic arguments and curve definition are presented as independent of the zero loci they describe.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The work assumes standard properties of modular forms and their q-expansions from number theory; the log Szegő curve is introduced as a new descriptive object but derived from the known Szegő curve.

axioms (1)
  • domain assumption Standard analytic properties of modular forms of weight k and their q-expansions hold for large k.
    Invoked for the asymptotic statements when k ≫ 0.
invented entities (1)
  • logarithmic version S_δ of the Szegő curve no independent evidence
    purpose: To describe the locus of zeros for intermediate δ
    Defined from the classical Szegő curve scaled by δ; no independent falsifiable prediction given beyond the zero locations themselves.

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Works this paper leans on

29 extracted references · 29 canonical work pages

  1. [1]

    preprint , pages=

    Orders in quadratic imaginary fields of small class number , author=. preprint , pages=

    work page

  2. [2]

    Watkins, Mark , TITLE =. Math. Comp. , FJOURNAL =. 2004 , NUMBER =. doi:10.1090/S0025-5718-03-01517-5 , URL =

    work page doi:10.1090/s0025-5718-03-01517-5 2004

  3. [3]

    1997 , PAGES =

    Serre, Jean-Pierre , TITLE =. 1997 , PAGES =. doi:10.1007/978-3-663-10632-6 , URL =

    work page doi:10.1007/978-3-663-10632-6 1997

  4. [4]

    Kohnen, Winfried , TITLE =. Comment. Math. Univ. St. Pauli , FJOURNAL =. 2003 , NUMBER =

    work page 2003

  5. [5]

    Brisebarre, Nicolas and Philibert, Georges , TITLE =. J. Ramanujan Math. Soc. , FJOURNAL =. 2005 , NUMBER =

    work page 2005

  6. [6]

    Szeg. \". Sitzungsberichte der Berliner Mathematischen Gesellschaft , year =

    work page

  7. [7]

    Discrete Math

    Gardy, Dani\`ele , TITLE =. Discrete Math. , FJOURNAL =. 1995 , NUMBER =. doi:10.1016/0012-365X(94)00133-4 , URL =

    work page doi:10.1016/0012-365x(94)00133-4 1995

  8. [8]

    Ramanujan J

    Heim, Bernhard and Neuhauser, Markus , TITLE =. Ramanujan J. , FJOURNAL =. 2022 , NUMBER =. doi:10.1007/s11139-022-00563-7 , URL =

    work page doi:10.1007/s11139-022-00563-7 2022

  9. [9]

    Mathematika , FJOURNAL =

    Rudnick, Ze\'ev , TITLE =. Mathematika , FJOURNAL =. 2024 , NUMBER =

    work page 2024

  10. [10]

    2024 , Eprint =

    Noam Kimmel , Title =. 2024 , Eprint =

    work page 2024

  11. [11]

    Raveh, Roei , TITLE =. Res. Number Theory , FJOURNAL =. 2025 , NUMBER =. doi:10.1007/s40993-025-00679-x , URL =

    work page doi:10.1007/s40993-025-00679-x 2025

  12. [12]

    Rudnick, Ze\'ev , TITLE =. Int. Math. Res. Not. , FJOURNAL =. 2005 , NUMBER =. doi:10.1155/IMRN.2005.2059 , URL =

    work page doi:10.1155/imrn.2005.2059 2005

  13. [13]

    Ghosh, Amit and Sarnak, Peter , TITLE =. J. Eur. Math. Soc. (JEMS) , FJOURNAL =. 2012 , NUMBER =. doi:10.4171/JEMS/308 , URL =

    work page doi:10.4171/jems/308 2012

  14. [14]

    and Jenkins, Paul , TITLE =

    Duke, W. and Jenkins, Paul , TITLE =. Pure Appl. Math. Q. , FJOURNAL =. 2008 , NUMBER =. doi:10.4310/PAMQ.2008.v4.n4.a15 , URL =

    work page doi:10.4310/pamq.2008.v4.n4.a15 2008

  15. [15]

    Rankin, F. K. C. and Swinnerton-Dyer, H. P. F. , TITLE =. Bull. London Math. Soc. , FJOURNAL =. 1970 , PAGES =. doi:10.1112/blms/2.2.169 , URL =

    work page doi:10.1112/blms/2.2.169 1970

  16. [16]

    Jennings-Shaffer, Chris and Swisher, Holly , TITLE =. RInt. J. Number Theory , FJOURNAL =. 2014 , NUMBER =

    work page 2014

  17. [17]

    Ramanujan Math

    Nicolas Brisebarre and Georges Philibert, Effective lower and upper bounds for the F ourier coefficients of powers of the modular invariant j , J. Ramanujan Math. Soc. 20 (2005), no. 4, 255--282. 2193216

    work page 2005

  18. [18]

    Duke and Paul Jenkins, On the zeros and coefficients of certain weakly holomorphic modular forms, Pure Appl

    W. Duke and Paul Jenkins, On the zeros and coefficients of certain weakly holomorphic modular forms, Pure Appl. Math. Q. 4 (2008), no. 4, 1327--1340. 2441704

    work page 2008

  19. [19]

    58 (2022), no

    Bernhard Heim and Markus Neuhauser, Asymptotic expansion of F ourier coefficients of reciprocals of E isenstein series , Ramanujan J. 58 (2022), no. 3, 871--887. 4441539

    work page 2022

  20. [20]

    Chris Jennings-Shaffer and Holly Swisher, A note on the transcendence of zeros of a certain family of weakly holomorphic forms, RInt. J. Number Theory 10 (2014), no. 2, 309--317

    work page 2014

  21. [21]

    Janis Klaise, Orders in quadratic imaginary fields of small class number, preprint (2012), 2

    work page 2012

  22. [22]

    Winfried Kohnen, Transcendence of zeros of E isenstein series and other modular functions , Comment. Math. Univ. St. Pauli 52 (2003), no. 1, 55--57. 1993952

    work page 2003

  23. [23]

    Number Theory 11 (2025), no

    Roei Raveh, On the zeros of the M iller basis of cusp forms , Res. Number Theory 11 (2025), no. 4, Paper No. 96, 32. 4978581

    work page 2025

  24. [24]

    F. K. C. Rankin and H. P. F. Swinnerton-Dyer, On the zeros of E isenstein series , Bull. London Math. Soc. 2 (1970), 169--170. 260674

    work page 1970

  25. [25]

    2, Paper No

    Ze\'ev Rudnick, Zeros of modular forms and F aber polynomials , Mathematika 70 (2024), no. 2, Paper No. e12244, 12. 4717387

    work page 2024

  26. [26]

    Vieweg & Sohn, Braunschweig, 1997, With a foreword by Brown and Serre

    Jean-Pierre Serre, Lectures on the M ordell- W eil theorem , third ed., Aspects of Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1997, With a foreword by Brown and Serre. 1757192

    work page 1997

  27. [27]

    G\' a bor Szeg o , \" U ber eine eigenschaft der exponentialreihe , Sitzungsberichte der Berliner Mathematischen Gesellschaft 23 (1924), 50--64

    work page 1924

  28. [28]

    The Sage Developers , S agemath, the S age M athematics S oftware S ystem ( V ersion 8.7) , 2019, https://www.sagemath.org

    work page 2019

  29. [29]

    Mark Watkins, Class numbers of imaginary quadratic fields, Math. Comp. 73 (2004), no. 246, 907--938. 2031415

    work page 2004