Distributions of reciprocal sums of parts in overpartitions and l‐regular partitions

Let be the sum of reciprocals of the parts in a partition . The study of partitions with conditions on was initiated by Graham, who showed that there is a partition of into distinct parts with for . In this paper, we study the distributions of in overpartitions and ‐regular partitions. Let be the set of overpartitions of . We prove that as , and Furthermore, for , we show that as , and where denotes the set of ‐regular partitions of . These results extend several striking findings from the celebrated paper by Kim and Kim.