For [Formula: see text], let [Formula: see text] denote the reciprocal sum up to x of numbers with k 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].
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Bayless et al. (2024) studied this question.
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