Higher Mertens constants for almost primes II

2023/03/11 by Bayless, Jonathan, Kinlaw, Paul, Lichtman, Jared Duker
#11A51 #11N25 #11N37 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2303.06481

Abstract

For k≥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 R₂(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 R₂(x) estimate. We also extend these results to R₃(x), and we remark on the general case k≥4.

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