Higher Mertens constants for almost primes. II. (Q6594430)

For any positive integer \(n\), let \(\Omega (n)\) denote the number of prime divisors of \(n\) (counted the multiplicity). Let \N\[\NR_k(x)= \sum_{n\le x, \Omega (n)=k} \frac 1n.\N\]\NIn [J. Number Theory 234, 448--475 (2022, Zbl 1497.11236)], the authors proved that \N\[\NR_2(x)= \frac 12(\log\log x+\beta )^2+\frac 12 (P(2)-\zeta (2))+\sum_{j=1}^{N-1} \frac{\alpha_j}{\log^j x}+O((\log x)^{-N}),\N\]\Nwhere (in the following, \(p\) denotes a prime) \N\[\N\beta =\gamma +\sum_p \left(\frac 1p +\log (1-\frac 1p)\right),\N\]\N\[\NP(2)=\sum_{p} \frac 1{p^2},\quad \zeta (2)=\sum_{n=1}^\infty \frac 1{n^2},\N\]\Nand \N\[\N\alpha_j=\lim_{y\to +\infty} \frac 1j\left( \frac{\log^j y}{j}-\sum_{p\le y} \frac{\log^j p}{p}\right), \quad j=1,2,\dots.\N\]\NIn this paper, the authors prove that for any \(m\ge 0\), \N\[\N\alpha_j=\frac{j! 2^j}{2j^2} (1+O(j^{-m})).\N\]\NThis confirm a conjecture in the above paper. In this paper, it also contains some results on \(R_3(x)\).