Higher Mertens constants for almost primes II

Abstract

For [Formula: see text], let [Formula: see text] denote the reciprocal sum up to [Formula: see text] of numbers with [Formula: see text] prime factors, counted with multiplicity. In prior work, the authors obtained estimates for [Formula: see text], extending Mertens’ second theorem, as well as a finer-scale estimate for [Formula: see text] up to [Formula: see text] error for any [Formula: see text]. In this paper, we establish the limiting behavior of the higher Mertens constants from the [Formula: see text] estimate. We also extend these results to [Formula: see text], and we comment on the general case [Formula: see text].