Higher Mertens constants for almost primes II

Abstract

For $k\ge 1$, 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 ${(logx)}^{-N}$ error for any $N>0$. In this paper, 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 comment on the general case $k\ge 4$.