Partition Frequency Moments: Modularity and Congruences
Hartosh Singh Bal
TL;DR
This work develops a universal Partition–Frequency framework that connects frequency moments of partition statistics to divisor sums and modular objects. By analyzing Euler-type products $A(q)$ and their companions $B(q)$, it shows that odd frequency moments yield half–integral weight modular forms on congruence subgroups and can be projected to arithmetic progressions with Sturm bounds certifying Ramanujan–type congruences; even moments land in the quasimodular arena. The authors certify several Ramanujan–type congruences for ordinary partitions (e.g., $M_3(7n+5)\equiv0\pmod7$ and $M_3(11n+6)\equiv0\pmod{11}$) and reveal a sharp contrast with overpartitions, where only zero–class congruences $M^{\overline{}}_m(\ell n)\equiv0\pmod{\ell}$ are observed in the scanned range. A Glaisher–character dictionary ties divisor filters to Dirichlet twists, enabling filtered congruences such as $\widehat{M}^{\chi_5}_3(5n+4)\equiv0\pmod{5}$, and connects these combinatorial statistics to classical modular objects like the $j$-invariant and Ramanujan’s $691$-prime phenomena. Together, the results establish a robust, certification-friendly pipeline for detecting and proving Ramanujan–type congruences across partition–type generating functions.
Abstract
We study frequency moments of partition statistics arising from Euler products $A(q)=\prod_{r\ge1}(1-q^r)^{-c(r)}$ via a transform that expresses the moment generating functions as $B(q)$ times explicit divisor--sum series determined by $c(r)$. When $A(q)$ is modular (typically an $η$--quotient), this yields (quasi)modular forms whose coefficients can be projected to arithmetic progressions and certified modulo primes by a Sturm bound, giving an effective pipeline for detecting and proving Ramanujan--type congruences for frequency moments. For ordinary partitions we recover and certify several congruences for odd moments in nonzero residue classes (e.g.\ $M_3(7n+5)\equiv 0\pmod7$ and $M_3(11n+6)\equiv 0\pmod{11}$). As a second input, we apply the same pipeline to overpartitions and certify a family of zero--class congruences $M_m^{\overline{\ }}(\ell n)\equiv 0\pmod{\ell}$ (including $m=5,7,11,13$), exhibiting a sharp contrast with the ordinary partition case: no nonzero residue--class congruences are observed for overpartition moments in our scan range. We also demonstrate that filtering the statistic via the Glaisher--character dictionary can itself create new Ramanujan--type progressions, e.g.\ a quadratic twist yields the certified congruence $\widehat{M}^{χ_5}_3(5n+4)\equiv 0\pmod{5}$.