Craig–van Ittersum–Ono conjecture on prime-detecting linear combinations of MacMahon’s M_a(n)

Determine whether, for each integer n, every n-linear combination of the MacMahon weighted partition numbers M_a(n) that detects primes (i.e., vanishes at n if and only if n is prime) is precisely an n-linear combination of the specific quasimodular forms listed in Craig–van Ittersum–Ono’s Table 1; equivalently, establish that an n-linear combination of the sequence M_a(n) is an n-linear combination of the quasimodular Eisenstein forms H_k (with k ≥ 6) if and only if it is an n-linear combination of the Table 1 forms of Craig–van Ittersum–Ono.

Background

The paper proves a classification of prime-detecting quasimodular forms: any quasimodular form whose nth Fourier coefficient vanishes if and only if n is prime must lie in the space of quasimodular Eisenstein series. In particular, Corollary 1.3 states that all such forms are linear combinations of D^n H_k for n ≥ 0 and k ≥ 6, where H_k are explicitly defined quasimodular Eisenstein series built from derivatives of Eisenstein series.

Craig, van Ittersum, and Ono introduced weighted partition functions M_a(n) via MacMahon’s q-series and showed that certain linear relations among M_a(n) detect primes. They conjectured a structural characterization: the n-linear combinations of M_a(n) that detect primes coincide with n-linear combinations of a specific list of quasimodular forms (their Table 1). Given the classification proved in this paper, the authors note that the original conjecture is equivalent to showing that an n-linear combination of M_a(n) lies in the span of H_k (k ≥ 6) if and only if it lies in the span of the forms in Craig–van Ittersum–Ono’s Table 1.