Distributions of reciprocal sums of parts in integer partitions

Abstract

Let $D_n$ be the set of partitions of n into distinct parts, and $\mathrm{srp}\left(\lambda \right)$ be the sum of reciprocals of the parts of the partition λ. We show that as $n\to \infty$,$E\left[\mathrm{srp}\right(\lambda ):\lambda \in D_n]\sim \frac{\log \left(3n\right)}{4}\text{and}\mathrm{Var}\left[\mathrm{srp}\right(\lambda ):\lambda \in D_n]\sim \frac{{\pi }^2}{24}.$ Moreover, for $P_n$, the set of ordinary partitions of n, we show that as $n\to \infty$,$E\left[\mathrm{srp}\right(\lambda ):\lambda \in P_n]\sim \pi \sqrt{\frac{n}{6}}\text{and}\mathrm{Var}\left[\mathrm{srp}\right(\lambda ):\lambda \in P_n]\sim \frac{{\pi }^2}{15}n.$ To prove these asymptotic formulas in a uniform manner, we derive a general asymptotic formula using Wright's circle method.