  For $k\ge1$, let $R_k(x)$ denote the reciprocal sum up to $x$ of numbers with
$k$ prime factors, counted with multiplicity. In prior work, the authors
obtained estimates for $R_k(x)$, extending Mertens' second theorem, as well as
a finer-scale estimate for $R_2(x)$ up to $(\log x)^{-N}$ error for any $N >
0$. In this article, we establish the limiting behavior of the higher Mertens
constants from the $R_2(x)$ estimate. We also extend these results to $R_3(x)$,
and we remark on the general case $k\ge4$.
