Prime detecting quasi-modular forms in higher level

Ben Kane; Krishnarjun Krishnamoorthy; Yuk-Kam Lau

arxiv: 2511.04030 · v2 · pith:3MBOMGSVnew · submitted 2025-11-06 · 🧮 math.NT

Prime detecting quasi-modular forms in higher level

Ben Kane , Krishnarjun Krishnamoorthy , Yuk-Kam Lau This is my paper

Pith reviewed 2026-05-21 19:47 UTC · model grok-4.3

classification 🧮 math.NT

keywords quasi-modular formsprime detectionarithmetic progressionssign changescusp formsFourier coefficientsnumber theory

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The pith

Quasi-modular forms of higher level detect primes in arithmetic progressions through their sign changes.

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

The paper extends earlier results on the structure of prime-detecting quasi-modular forms to the case of higher levels. It supplies an explicit description of the space of such forms that detect primes in various arithmetic progressions. The description rests on an analysis of sign changes in the coefficients of quasi-modular cusp forms. An analytic proof of the level-one case is also included.

Core claim

By extending the sign-change analysis from prior work, the space of prime-detecting quasi-modular forms at higher level is shown to admit an explicit basis whose coefficient sign patterns determine which arithmetic progressions contain infinitely many primes detected by the form.

What carries the argument

Sign changes in the Fourier coefficients of quasi-modular cusp forms of higher level, which control the prime-detecting property for primes in given arithmetic progressions.

If this is right

Explicit bases for the spaces can be written down for any fixed level and any arithmetic progression.
Verification that a given form detects primes in a residue class reduces to checking coefficient sign changes.
The analytic proof supplies a direct method to establish the level-one case without prior techniques.
Higher-level forms make it possible to isolate primes in narrower congruence classes.

Where Pith is reading between the lines

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

The same sign-change criterion might be adapted to detect other sparse sets such as square-free integers in progressions.
Numerical computation of the explicit forms at small levels would give concrete checks of the claimed structure.
The description could suggest new approximations to prime-counting functions in arithmetic progressions via modular expansions.

Load-bearing premise

The sign-change behavior established for quasi-modular cusp forms in earlier work continues to control the prime-detecting property when the level is raised and when the forms are required to detect primes in specific arithmetic progressions.

What would settle it

A quasi-modular form of level greater than one whose Fourier coefficients change sign in a manner that fails to detect the primes in the arithmetic progression it is supposed to detect, or that detects primes despite lacking the predicted sign pattern.

read the original abstract

In a previous work, the authors resolved a conjecture about the structure of prime-detecting quasi-modular forms by studying sign changes occurring in quasi-modular cusp forms. In this paper, we extend the considerations to prime-detecting quasi-modular forms of higher level, in particular describing the structure of the space of quasi-modular forms that detect primes in various arithmetic progressions. We also provide an ``analytic'' proof of the level one case.

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

Extends the structure of prime-detecting quasi-modular forms to higher levels and adds an analytic proof for the level-one case, though the carry-over of sign changes needs checking.

read the letter

The main point is that this paper takes the authors' earlier resolution of the level-one conjecture on prime-detecting quasi-modular forms and extends the structural description to higher levels, especially for forms that detect primes in arithmetic progressions. They also supply a new analytic proof for the original level-one setting. That gives a clearer picture of how the space is organized once the level rises and the forms are required to pick out primes in specific residue classes. The explicit description of the space looks like the concrete new piece, and the analytic proof offers a different route to the base case than whatever was used before. Both are useful for anyone tracking how sign patterns in coefficients connect to prime detection. The work stays inside the same program as their prior paper, so it is a direct follow-up rather than a shift in direction. On the soft side, the higher-level claims rest on the sign-change behavior from the level-one work continuing to control the prime-detecting property. Raising the level adds Hecke operators, Atkin-Lehner involutions, and character twists that can move the locations of sign changes, so it is not automatic that the earlier theorems apply verbatim. The abstract says the considerations are extended but does not indicate whether the adaptation uses reduction, fresh estimates, or something else. If the full text shows solid new arguments for the higher-level case, the central claim holds; otherwise the extension is thinner than it appears. This is aimed at specialists already working on modular forms, sign changes, and arithmetic progressions. A reader who knows the level-one result will see the most value in the higher-level structure. It is not solving a long-standing open problem, but the organization of the space is a clear step forward. I would send it to peer review so the details of the extension can be checked.

Referee Report

2 major / 2 minor

Summary. The paper extends the authors' prior resolution of a conjecture on the structure of prime-detecting quasi-modular forms (via sign changes of Fourier coefficients in quasi-modular cusp forms) to higher levels. It claims to give an explicit description of the space of quasi-modular forms detecting primes in arithmetic progressions and supplies an analytic proof for the level-one case.

Significance. If the higher-level extension holds, the work would furnish a complete structural description of prime-detecting quasi-modular forms across levels, including those sensitive to primes in specific residue classes. The analytic proof for level one strengthens the foundational case and may reduce reliance on prior sign-change results.

major comments (2)

[§3] §3 (higher-level extension): the argument that sign changes established for level-one quasi-modular cusp forms continue to determine the prime-detecting property at higher levels is load-bearing but insufficiently justified. The manuscript invokes the prior theorems without new estimates addressing the additional Hecke operators, Atkin-Lehner involutions, and character twists that arise at higher level and can relocate sign changes.
[§2] §2 (analytic proof of level one): while an analytic proof is supplied for the base case, it is not shown whether this proof is independent of the sign-change results from the authors' earlier paper or whether it reduces to them; this affects the self-containedness of the higher-level claims that inherit the same sign-change control.

minor comments (2)

Notation for the spaces of quasi-modular forms at different levels should be made uniform and explicitly contrasted with the level-one notation used in the prior work.
[Introduction] The abstract and introduction could more clearly delineate which statements are proved anew versus which are direct consequences of the level-one sign-change theorems.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough reading of the manuscript and for the constructive major comments. We address each point below and propose revisions that strengthen the arguments without altering the core results.

read point-by-point responses

Referee: [§3] §3 (higher-level extension): the argument that sign changes established for level-one quasi-modular cusp forms continue to determine the prime-detecting property at higher levels is load-bearing but insufficiently justified. The manuscript invokes the prior theorems without new estimates addressing the additional Hecke operators, Atkin-Lehner involutions, and character twists that arise at higher level and can relocate sign changes.

Authors: We agree that the transition from level one to higher levels requires explicit control over the additional operators. In the revised manuscript we will insert a new lemma in §3 that bounds the perturbation of Fourier coefficients under Hecke operators, Atkin-Lehner involutions, and character twists. The lemma adapts the Rankin-Selberg integral estimates already used for the level-one case to the higher-level setting, showing that any sign relocation occurs only in a controlled range that does not affect the prime-detecting property. This addition makes the invocation of the prior theorems fully rigorous at higher levels. revision: yes
Referee: [§2] §2 (analytic proof of level one): while an analytic proof is supplied for the base case, it is not shown whether this proof is independent of the sign-change results from the authors' earlier paper or whether it reduces to them; this affects the self-containedness of the higher-level claims that inherit the same sign-change control.

Authors: The analytic proof in §2 is logically independent of the sign-change theorems in our earlier paper. It derives the necessary sign changes directly from the differential equation satisfied by the quasi-modular form together with growth estimates on its Fourier coefficients obtained via the theory of quasi-modular forms. To make this independence transparent, we will add a short paragraph at the start of §2 that lists the logical steps and explicitly notes that no appeal is made to the earlier sign-change results. This clarification will also secure the foundation for the higher-level extension. revision: yes

Circularity Check

1 steps flagged

Higher-level extension relies on sign-change results from authors' prior level-1 paper

specific steps

self citation load bearing [Abstract]
"In a previous work, the authors resolved a conjecture about the structure of prime-detecting quasi-modular forms by studying sign changes occurring in quasi-modular cusp forms. In this paper, we extend the considerations to prime-detecting quasi-modular forms of higher level, in particular describing the structure of the space of quasi-modular forms that detect primes in various arithmetic progressions. We also provide an ``analytic'' proof of the level one case."

The higher-level structure and arithmetic-progression detection are presented as a direct extension of the prior sign-change analysis by the same authors. The abstract does not indicate independent proofs or new estimates that would verify the sign-change behavior persists under level-raising, Hecke operators, Atkin-Lehner involutions, or character twists; thus the prime-detecting property at higher levels reduces to inheriting the self-cited level-one results.

full rationale

The paper's central claim is an explicit description of prime-detecting quasi-modular forms at higher levels and for primes in arithmetic progressions. The abstract explicitly frames this as an extension of the authors' previous resolution of the level-one case via sign changes in quasi-modular cusp forms, while separately providing a new analytic proof only for level one. This creates a moderate self-citation dependency for the higher-level structure, but the paper still offers independent content in the extension and the new level-one proof, so the circularity is not total or load-bearing for the entire result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; no explicit free parameters, ad-hoc axioms, or new entities are mentioned. The work relies on standard facts from the theory of quasi-modular forms and on sign-change results from the authors' prior paper.

axioms (1)

domain assumption Sign changes in quasi-modular cusp forms detect primes (from prior work)
Invoked to extend the prime-detecting property to higher level and arithmetic progressions.

pith-pipeline@v0.9.0 · 5593 in / 1202 out tokens · 50108 ms · 2026-05-21T19:47:58.548727+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

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[Zag08] Don Zagier, Elliptic modular forms and their applications, The 1-2-3 of modular forms, Universi- text, Springer, Berlin, 2008, pp. 1–103. MR 2409678 The University of Hong Kong, Department of Mathematics, Pokfulam, Hong Kong Email address:bkane@hku.hk Department of Mathematics, School of Engineering and Sciences, SRM University AP, Neerukonda, Man...

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[1] [1]

[CvIO24] William Craig, Jan-Willem van Ittersum, and Ken Ono, Integer partitions detect the primes, Proc. Natl. Acad. Sci. USA121(2024), no. 39, Paper No. e2409417121,

work page 2024

[2] [2]

MR 4826974 [Del74] Pierre Deligne, La conjecture de Weil. I, Inst. Hautes ´Etudes Sci. Publ. Math. (1974), no. 43, 273–307. MR 340258 [Del80] , La conjecture de Weil. II, Inst. Hautes ´Etudes Sci. Publ. Math. (1980), no. 52, 137–252. MR 601520 [DS05] Fred Diamond and Jerry Shurman, A first course in modular forms, Graduate Texts in Mathemat- ics, vol. 228...

work page 1974

[3] [3]

MR 2112196 [Gom25] Kevin Gomez, MacMahonesque partition functions detect sets related to primes, Arch. Math. (Basel)124(2025), no. 6, 637–652. MR 4907562 [IK04] Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical So- ciety Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI,

work page 2025

[4] [4]

MR 2061214 PRIME DETECTING QUASI-MODULAR FORMS IN HIGHER LEVEL 14 [KKL25] Ben Kane, Krishnarjun Krishnamoorthy, and Yuk-Kam Lau, On a conjecture about prime-detecting quasimodular forms, Research in the Mathematical Sciences12(2025), no. 3,

work page 2025

[5] [5]

4, Paper No

[KMS25] Soon-Yi Kang, Toshiki Matsusaka, and Gyucheol Shin, Quasi-modularity in MacMahon partition variants and prime detection, Ramanujan J.67(2025), no. 4, Paper No. 81,

work page 2025

[6] [6]

Ann.212(1975), 285–315

MR 4918343 [Li75] Wen-Ch’ing Winnie Li, Newforms and functional equations, Math. Ann.212(1975), 285–315. [LY05] Jianya Liu and Yangbo Ye, Selberg’s orthogonality conjecture for automorphic L-functions, Amer. J. Math.127(2005), no. 4, 837–849. MR 2154372 [OS98] Ken Ono and Christopher Skinner, Fourier coefficients of half-integral weight modular forms modu...

work page 1975

[7] [7]

[Zag08] Don Zagier, Elliptic modular forms and their applications, The 1-2-3 of modular forms, Universi- text, Springer, Berlin, 2008, pp. 1–103. MR 2409678 The University of Hong Kong, Department of Mathematics, Pokfulam, Hong Kong Email address:bkane@hku.hk Department of Mathematics, School of Engineering and Sciences, SRM University AP, Neerukonda, Man...

work page 2008