Partitions with prescribed sum of reciprocals: asymptotic bounds

Abstract

In $1963$ Graham proved that every positive integer $n\ge 78$ can be written as a sum of distinct positive integers $a_1,a_2,\dots ,a_r$ for which $\frac{1}{a_1}+\frac{1}{a_2}+\dots +\frac{1}{a_r}$ is equal to $1$. In the same paper he managed to further generalize this, and showed that for all positive rationals $\alpha$ and all positive integers $m$, there exists an $n_{\alpha ,m}$ such that every positive integer $n\ge n_{\alpha ,m}$ has a partition with distinct parts, all larger than or equal to $m$, and such that the sum of reciprocals is equal to $\alpha$. No attempt was made to estimate the quantity $n_{\alpha ,m}$, however. With $n_{\alpha }:=n_{\alpha ,1}$, in this paper we provide near-optimal upper bounds on $n_{\alpha }$ and $n_{\alpha ,m}$, as well as bounds on the cardinality of the set $\{\alpha :n_{\alpha }\le n\}$.

12 pages