Distributions of reciprocal sums of parts in integer partitions

Abstract

Let Dn be the set of partitions of n into distinct parts, and srp(λ) be the sum of reciprocals of the parts of the partition λ. We show that as n→∞, E[srp(λ):λ∈Dn]∼[Formula presented]. Moreover, for Pn, the set of ordinary partitions of n, we show that as n→∞, E[srp(λ):λ∈Pn]∼π[Formula presented]andVar[srp(λ):λ∈Pn]∼[Formula presented]n. To prove these asymptotic formulas in a uniform manner, we derive a general asymptotic formula using Wright's circle method. © 2024 Elsevier Inc.