MacMahonesque partition functions detect sets related to primes
Kevin Gomez
TL;DR
The paper extends the prime-detection program of MacMahon-type partition functions by defining MacMahonesque functions $M_{\vec{a}}(n)$ and their generating series, and formalizes detection of sets via $q$-series using $a_n=0$ for $n\in S$. It constructs infinite families of linearly independent detectors for prime cubes through quasimodular methods, notably via $g_{k,\ell}$ built from derivatives of Eisenstein series, and then expresses these detectors as combinations of the $\mathcal{U}_{\vec{a}}(q)$, illustrating weight-raising by the differential operator $D$. The work further shows how twists by roots of unity yield detectors for primes in arithmetic progressions, by introducing the twisted quasimodular forms $G_k^{r,t}$ and the corresponding $f_{k,\ell}^{r,t}$, which strongly detect primes in residue classes modulo $t$. Collectively, these results fuse partition theory with quasimodular forms to produce new, infinite families of prime-detecting identities and reveal a structural link between modular-type objects and prime-set detection in a combinatorial generating-function framework.
Abstract
Recent work by Craig, van Ittersum, and Ono constructs explicit expressions in the partition functions of MacMahon that detect the prime numbers. Furthermore, they define generalizations, the MacMahonesque functions, and prove there are infinitely many such expressions in these functions. Here, we show how to modify and adapt their construction to detect cubes of primes as well as primes in arithmetic progressions.