Higher Mertens constants for almost primes

For $k\ge1$, a $k$-almost prime is a positive integer with exactly $k$ prime factors, counted with multiplicity. In this article we give elementary proofs of precise asymptotics for the reciprocal sum of $k$-almost primes. Our results match the strength of those of classical analytic methods. We also study the limiting behavior of the constants appearing in these estimates, which may be viewed as higher analogues of the Mertens constant $β=0.2614...$ Further, in the case $k=2$ of semiprimes we give yet finer-scale and explicit estimates, as well as a conjecture.

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