Asymptotics of reciprocal supernorm partition statistics

We consider two multiplicative statistics on the set of integer partitions: the norm of a partition, which is the product of its parts, and the supernorm of a partition, which is the product of the prime numbers \(p_i\) indexed by its parts i. We introduce and study new statistics that are sums of reciprocals of supernorms on three statistical ensembles of partitions, labelled by their size \(|\lambda |=n\), their perimeter equaling n, and their largest part equaling n. We show that the cumulative statistics of the reciprocal supernorm for each of the three ensembles are asymptotic to \(e^{\gamma } \log n\) as \(n \rightarrow \infty \).