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arXiv · 2507.20432

Quasimodular forms that detect primes are Eisenstein

Jan-Willem van Ittersum·Lukas Mauth·Ken Ono·Ajit Singh
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Abstract

MacMahon's partition functions and their extensions provide equations that identify prime numbers as solutions. These results depend on the theory of (mixed weight) quasimodular forms on $SL_2(\mathbb{Z})$. Two of the authors, along with Craig, conjectured an explicit description of the set of prime-detecting quasimodular forms in terms of Eisenstein series and their derivatives. Kane et al.\ recently verified this conjecture using analytic methods. We offer an alternative proof using the theory of $\ell$-adic Galois representations associated to modular forms.

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BibTeXRIS

Jan-Willem van Ittersum, Lukas Mauth, Ken Ono, Ajit Singh. 2025-07-27. Quasimodular forms that detect primes are Eisenstein. https://arxiv.org/abs/2507.20432

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