Partition Frequency Moments: Modularity and Congruences

Abstract

We study frequency moments of partition statistics arising from Euler products $A(q)={\prod }_{r\ge 1}(1-q^r{)}^{-c(r)}$ 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 $\hat{I}\cdot$--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.\ $M_3(7n+5)\equiv 0(\mathrm{mod}7)$ and $M_3(11n+6)\equiv 0(\mathrm{mod}11)$). As a second input, we apply the same pipeline to overpartitions and certify a family of zero--class congruences $M_m^{\overline{}}(\ell n)\equiv 0(\mathrm{mod}\ell )$ (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 ${\hat{M}}_3^{\ddot{I}{‡}_5}(5n+4)\equiv 0(\mathrm{mod}5)$.