Quasimodular forms that detect primes are Eisenstein

2025/07/27 by van Ittersum, Jan-Willem, Mauth, Lukas, Ono, Ken +1 · 1 citation
#05A17 #11Fxx #11P81 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.20432

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 SL₂(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 ℓ-adic Galois representations associated to modular forms.

Citations

• On a conjecture about prime-detecting quasimodular forms
• Quasi-modularity in MacMahon partition variants and prime detection
• MacMahonesque partition functions detect sets related to primes
• Higher level q-multiple zeta values with applications to quasimodular forms and partitions
• Integer partitions detect the primes
• Propri�t�s galoisiennes des points d'ordre fini des courbes elliptiques

Cited by

• Prime detecting quasi-modular forms in higher level
• On a conjecture about prime-detecting quasimodular forms

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• On a conjecture about prime-detecting quasimodular forms
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