Zeros of certain combinations of Eisenstein series of weight 2k, 3k, and k + l

Jetjaroen Klangwang

arxiv: 1907.04259 · v1 · pith:VS5WKPIVnew · submitted 2019-07-09 · 🧮 math.NT

Zeros of certain combinations of Eisenstein series of weight 2k, 3k, and k + l

Jetjaroen Klangwang This is my paper

Pith reviewed 2026-05-25 00:04 UTC · model grok-4.3

classification 🧮 math.NT

keywords Eisenstein serieszerosmodular formsfundamental domainweightlinear combinationsSL(2,Z)

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The pith

For sufficiently large k and l, all zeros of E_k² + E_{2k}, E_k³ + E_{3k}, and E_k E_l + E_{k+l} lie on the arc A in the fundamental domain.

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

The paper establishes the location of zeros for three families of modular forms built from Eisenstein series. It shows that when the weights are large enough, these zeros sit exclusively on the specified boundary arc. This builds directly on prior results for single Eisenstein series by Rankin and Swinnerton-Dyer. A reader would care because it pins down the vanishing behavior of these forms without computing each case separately. The result applies uniformly to the listed combinations.

Core claim

By utilizing work of F.K.C. Rankin and Swinnerton-Dyer, the paper proves that for sufficiently large k,l, all zeros in the standard fundamental domain of E_k²(τ) + E_{2k}(τ), E_k³(τ) + E_{3k}(τ), and E_k(τ)E_l(τ) + E_{k+l}(τ) are located on the lower boundary A = { e^{iθ} : π/2 ≤ θ ≤ 2π/3 }.

What carries the argument

Extension of zero-location results from individual Eisenstein series E_m(τ) to the linear combinations E_k² + E_{2k} etc. for large weights.

If this is right

The three combinations have all their zeros on the arc A when k and l are sufficiently large.
No zeros occur in the interior or on other boundaries of the fundamental domain for these forms under the same condition.
The location result holds for each of the three specific combinations listed.

Where Pith is reading between the lines

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

If the extension holds, then for even larger weights the combinations behave increasingly like the highest weight term.
One could test the result computationally for increasing k to estimate the threshold where 'sufficiently large' begins.
This approach might extend to other linear combinations or to higher level modular groups.

Load-bearing premise

The zero location results for individual Eisenstein series extend directly to these linear combinations when the weights k and l are sufficiently large.

What would settle it

Finding a zero of any of the three forms that lies in the fundamental domain but off the arc A, for some pair of sufficiently large k and l.

read the original abstract

We locate the zeros of the modular forms $E_k^2(\tau) + E_{2k}(\tau), E_k^3(\tau) + E_{3k} (\tau),$ and $E_k(\tau)E_l(\tau) +E_{k+l}(\tau),$ where $E_k(\tau)$ is the Eisenstein series for the full modular group $\text{SL}_2(\mathbb{Z})$. By utilizing work of F.K.C. Rankin and Swinnerton-Dyer, we prove that for sufficiently large $k,l$, all zeros in the standard fundamental domain are located on the lower boundary $\mathcal{A} = \{ e^{i\theta} : \pi/2 \leq \theta \leq 2\pi/3\}$.

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.

Desk Editor's Note private letter to a colleague

Short note applies Rankin-Swinnerton-Dyer zero-location to three new Eisenstein combinations; the key step is whether the sums inherit the arc property without extra cancellation control.

read the letter

The paper proves that for large enough k and l the zeros of E_k² + E_{2k}, E_k³ + E_{3k}, and E_k E_l + E_{k+l} all lie on the arc A inside the fundamental domain. It does this by invoking the Rankin-Swinnerton-Dyer results on single Eisenstein series and claiming the same location holds for these linear combinations. That is the concrete new statement; the combinations themselves had not been treated before in the cited literature. The work is a straightforward extension of known techniques to a few specific forms that happen to be closed under the relevant operations. If the adaptation is carried through with the necessary estimates, the claim is clean and the argument is short. The potential soft spot is exactly the one the stress-test flags: the original proofs rely on the Fourier expansion and growth of a single E_m, and sums can in principle cancel off the arc even when each term is nonzero. The abstract only says the results are used, so the manuscript must supply a separate uniform estimate showing that one term dominates or that cancellation cannot produce extra zeros for large weights. If that quantitative step is present and correct, the paper is fine; if it is only sketched, the central claim rests on an unverified transfer. This is niche modular-forms work aimed at people already following zero-location results for Eisenstein series. A reader who wants to see the classical arc property preserved under these particular sums will find it useful. It is narrow enough that it does not reorganize the field, but the statement is precise and the method is standard, so it deserves a referee to check the details of the adaptation rather than a desk reject.

Referee Report

1 major / 0 minor

Summary. The paper claims that, by utilizing results of F.K.C. Rankin and Swinnerton-Dyer on the location of zeros of individual Eisenstein series E_m for large even m, the zeros of the three linear combinations E_k²(τ) + E_{2k}(τ), E_k³(τ) + E_{3k}(τ), and E_k(τ)E_l(τ) + E_{k+l}(τ) all lie on the arc A = {e^{iθ} : π/2 ≤ θ ≤ 2π/3} inside the standard fundamental domain, for all sufficiently large k and l.

Significance. If the transfer of the Rankin-Swinnerton-Dyer non-vanishing statements to these specific combinations can be made rigorous, the result would extend classical zero-location theorems from single Eisenstein series to a natural family of combinations, with potential consequences for the study of zero distributions of modular forms built from Eisenstein series.

major comments (1)

The central claim requires a separate argument that the non-vanishing off A established by Rankin-Swinnerton-Dyer for single E_m carries over to the listed linear combinations without off-arc cancellation. The abstract states only that the result follows “by utilizing work of” those authors; no quantitative dominance estimate or modified positivity argument ruling out cancellation for |τ| > 1 is supplied in the provided text. This adaptation is load-bearing for the proof of the stated location theorem.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need for greater explicitness in adapting the Rankin-Swinnerton-Dyer results. We address the major comment below and will revise the paper to strengthen the exposition.

read point-by-point responses

Referee: The central claim requires a separate argument that the non-vanishing off A established by Rankin-Swinnerton-Dyer for single E_m carries over to the listed linear combinations without off-arc cancellation. The abstract states only that the result follows “by utilizing work of” those authors; no quantitative dominance estimate or modified positivity argument ruling out cancellation for |τ| > 1 is supplied in the provided text. This adaptation is load-bearing for the proof of the stated location theorem.

Authors: We agree that an explicit argument ruling out off-arc cancellation in the combinations is necessary and that the current text relies on the reader to infer the adaptation from the cited works. In the revised manuscript we will insert a new subsection (immediately following the statement of the main theorem) that supplies the required quantitative estimates. Using the asymptotic expansions and sign properties established by Rankin and Swinnerton-Dyer for |τ| > 1, we show that |E_{2k}(τ)| = o(|E_k(τ)|^2) uniformly off A for large even k; the same comparison holds, with adjusted constants, for the cubic and mixed cases. These bounds preclude cancellation outside A and thereby transfer the non-vanishing statements directly to the three forms. The revision will therefore make the proof self-contained while preserving the original logical structure. revision: yes

Circularity Check

0 steps flagged

No circularity detected; central claim rests on external cited results

full rationale

The paper's derivation explicitly invokes the zero-location theorems of Rankin and Swinnerton-Dyer (distinct authors) for individual Eisenstein series and claims an extension to the listed linear combinations for large k,l. No self-citation appears in the abstract or described chain, no parameter is fitted then relabeled as prediction, and no ansatz or uniqueness result is imported from the present author's prior work. The argument is therefore not self-referential by construction; any question of whether the extension is rigorously justified belongs to correctness rather than circularity analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard properties of Eisenstein series and external theorems; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

standard math Eisenstein series E_m(τ) are modular forms of weight m for SL_2(Z)
Used to define the combinations whose zeros are studied.
domain assumption Rankin-Swinnerton-Dyer theorems locate zeros of individual Eisenstein series on the arc A for large weight
Invoked to transfer the location result to the listed combinations for large k and l.

pith-pipeline@v0.9.0 · 5667 in / 1295 out tokens · 22113 ms · 2026-05-25T00:04:09.567007+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · 1 internal anchor

[1]

Wohlfahrt, ¨Uber die nullstellen einiger eisensteinreihen, Mathematische Nachrichten 26 (6) (1963) 381–383

K. Wohlfahrt, ¨Uber die nullstellen einiger eisensteinreihen, Mathematische Nachrichten 26 (6) (1963) 381–383

work page 1963
[2]

R. A. Rankin, The Zeros of Eisenstein Series: Dedicated to the Me mory of Professor Ananda Rau, Ramanujan Institute, 1969

work page 1969
[3]

F. K. C. Rankin, H. P. F. Swinnerton-Dyer, On the zeros of eisen stein series, Bulletin of the London Mathematical Society 2 (2) (1970) 16 9–170

work page 1970
[4]

Miezaki, H

T. Miezaki, H. Nozaki, J. Shigezumi, On the zeros of eisenstein ser ies for Γ∗ 0(2) and Γ ∗ 0(3), Journal of the Mathematical Society of Japan 59 (3) (2007) 693–706

work page 2007
[5]

Shigezumi, On the zeros of the eisenstein series for Γ ∗ 0(5) and Γ ∗ 0(7), Kyushu Journal of Mathematics 61 (2) (2007) 527–549

J. Shigezumi, On the zeros of the eisenstein series for Γ ∗ 0(5) and Γ ∗ 0(7), Kyushu Journal of Mathematics 61 (2) (2007) 527–549

work page 2007
[6]

Garthwaite, L

S. Garthwaite, L. Long, H. Swisher, S. Treneer, Zeros of some level 2 eisen- stein series, Proceedings of The American Mathematical Society 13 8 (2) (2010) 467–480

work page 2010
[7]

Hahn, On zeros of eisenstein series for genus zero fuchsian g roups, Pro- ceedings of the American Mathematical Society 135 (8) (2007) 239 1–2401

H. Hahn, On zeros of eisenstein series for genus zero fuchsian g roups, Pro- ceedings of the American Mathematical Society 135 (8) (2007) 239 1–2401

work page 2007
[8]

Getz, A generalization of a theorem of Rankin and Swinnerton- Dyer on zeros of modular forms, Proceedings of the American Mathematica l Society 132 (8) (2004) 2221–2231

J. Getz, A generalization of a theorem of Rankin and Swinnerton- Dyer on zeros of modular forms, Proceedings of the American Mathematica l Society 132 (8) (2004) 2221–2231

work page 2004
[9]

Gun, On the zeros of certain cusp forms, Mathematical Proc eedings of the Cambridge Philosophical Society 141 (2) (2006) 191–195

S. Gun, On the zeros of certain cusp forms, Mathematical Proc eedings of the Cambridge Philosophical Society 141 (2) (2006) 191–195

work page 2006
[10]

W. Duke, P. Jenkins, On the zeros and coeﬃcients of certain we akly holo- morphic modular forms, Pure and Applied Mathematics Quarterly 4 (4 ) (2008) 1327–1340. 17

work page 2008
[11]

Zeros of weakly holomorphic modular forms of levels 2 and 3

S. Garthwaite, P. Jenkins, Zeros of weakly holomorphic modular forms of levels 2 and 3, arXiv preprint arXiv:1205.7050

work page internal anchor Pith review Pith/arXiv arXiv
[12]

Haddock, P

A. Haddock, P. Jenkins, Zeros of weakly holomorphic modular fo rms of level 4, International Journal of Number Theory 10 (02) (2014) 455–470

work page 2014
[13]

Reitzes, P

S. Reitzes, P. Vulakh, M. P. Young, Zeros of certain combinatio ns of eisen- stein series, Mathematika 63 (2) (2017) 666–695. 18

work page 2017

[1] [1]

Wohlfahrt, ¨Uber die nullstellen einiger eisensteinreihen, Mathematische Nachrichten 26 (6) (1963) 381–383

K. Wohlfahrt, ¨Uber die nullstellen einiger eisensteinreihen, Mathematische Nachrichten 26 (6) (1963) 381–383

work page 1963

[2] [2]

R. A. Rankin, The Zeros of Eisenstein Series: Dedicated to the Me mory of Professor Ananda Rau, Ramanujan Institute, 1969

work page 1969

[3] [3]

F. K. C. Rankin, H. P. F. Swinnerton-Dyer, On the zeros of eisen stein series, Bulletin of the London Mathematical Society 2 (2) (1970) 16 9–170

work page 1970

[4] [4]

Miezaki, H

T. Miezaki, H. Nozaki, J. Shigezumi, On the zeros of eisenstein ser ies for Γ∗ 0(2) and Γ ∗ 0(3), Journal of the Mathematical Society of Japan 59 (3) (2007) 693–706

work page 2007

[5] [5]

Shigezumi, On the zeros of the eisenstein series for Γ ∗ 0(5) and Γ ∗ 0(7), Kyushu Journal of Mathematics 61 (2) (2007) 527–549

J. Shigezumi, On the zeros of the eisenstein series for Γ ∗ 0(5) and Γ ∗ 0(7), Kyushu Journal of Mathematics 61 (2) (2007) 527–549

work page 2007

[6] [6]

Garthwaite, L

S. Garthwaite, L. Long, H. Swisher, S. Treneer, Zeros of some level 2 eisen- stein series, Proceedings of The American Mathematical Society 13 8 (2) (2010) 467–480

work page 2010

[7] [7]

Hahn, On zeros of eisenstein series for genus zero fuchsian g roups, Pro- ceedings of the American Mathematical Society 135 (8) (2007) 239 1–2401

H. Hahn, On zeros of eisenstein series for genus zero fuchsian g roups, Pro- ceedings of the American Mathematical Society 135 (8) (2007) 239 1–2401

work page 2007

[8] [8]

Getz, A generalization of a theorem of Rankin and Swinnerton- Dyer on zeros of modular forms, Proceedings of the American Mathematica l Society 132 (8) (2004) 2221–2231

J. Getz, A generalization of a theorem of Rankin and Swinnerton- Dyer on zeros of modular forms, Proceedings of the American Mathematica l Society 132 (8) (2004) 2221–2231

work page 2004

[9] [9]

Gun, On the zeros of certain cusp forms, Mathematical Proc eedings of the Cambridge Philosophical Society 141 (2) (2006) 191–195

S. Gun, On the zeros of certain cusp forms, Mathematical Proc eedings of the Cambridge Philosophical Society 141 (2) (2006) 191–195

work page 2006

[10] [10]

W. Duke, P. Jenkins, On the zeros and coeﬃcients of certain we akly holo- morphic modular forms, Pure and Applied Mathematics Quarterly 4 (4 ) (2008) 1327–1340. 17

work page 2008

[11] [11]

Zeros of weakly holomorphic modular forms of levels 2 and 3

S. Garthwaite, P. Jenkins, Zeros of weakly holomorphic modular forms of levels 2 and 3, arXiv preprint arXiv:1205.7050

work page internal anchor Pith review Pith/arXiv arXiv

[12] [12]

Haddock, P

A. Haddock, P. Jenkins, Zeros of weakly holomorphic modular fo rms of level 4, International Journal of Number Theory 10 (02) (2014) 455–470

work page 2014

[13] [13]

Reitzes, P

S. Reitzes, P. Vulakh, M. P. Young, Zeros of certain combinatio ns of eisen- stein series, Mathematika 63 (2) (2017) 666–695. 18

work page 2017