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Source. Proposition 4.1, p. 17, of Ofir Gorodetsky, Jared Duker Lichtman and Mo Dick Wong, On Erdős sums of almost primes, C. R. Math. Acad. Sci. Paris 362 (2024), 1571--1596, doi:10.5802/crmath.650, as named on the source card; labels and pages are those of arXiv:2303.08277v2 (12 May 2024).

Statement

Setting (p. 1). $f_k=\sum_{\Omega(n)=k}1/(n\log n)$, with $\Omega(n)$ the number of prime factors of $n$ counted with multiplicity.

Proposition 4.1 (p. 17, quoted). "We have $f_k=1+O(k/2^{k/4})$."

The authors say that, in view of Theorem 1.2, they did not try to optimise the exponent (p. 17), and that the argument has potentially much wider applicability to primitive sets other than the $k$-almost primes (p. 3).

Read depth. Claims checked: the statement and the outline of Section 4 were read on pp. 17--23. The lemmas were not checked step by step. Nothing here is independently reviewed.

Proof pointer

Section 4.2, pp. 20--23. With $j=\lfloor k/4\rfloor$ and $y=2^j$, the $n$ whose $(j+1)$-th largest prime factor is below $e^y$ contribute $O(k/2^{k/4})$ (Corollary 4.6, p. 19, from a bound of Erdős and Sárközy, Lemma 4.5). For the rest, Mertens' theorem turns $1/(n\log n)$ into $e^\gamma$ times $\log P_1(n)/\log n$ times the density of the integers $bn$ whose prime factors in $b$ are all at least the largest prime factor $P_1(n)$ of $n$ (the paper's (4.7)). Writing $\log n/\log P_1(n)$ as a nested expression in the ratios $\log P_{i+1}(n)/\log P_i(n)$, and showing that these ratios behave like independent uniform variables (Lemmas 4.3, 4.4 and 4.7), bounds the sum above by $(e^\gamma+O(k/2^{k/4}))I_j(1)$ and below by $(e^\gamma-O(k/2^{k/4}))I_j(k-j)$ (the paper's (4.18) and (4.19)). Theorem 4.8 (p. 22; see Theorem 1.6) evaluates both integrals as $e^{-\gamma}+O(k/2^j)$.

Dependencies

Theorem 1.6 in its general form, Theorem 4.8 (p. 22). Within the paper: Lemmas 4.2--4.5 and Corollary 4.6 (pp. 17--19), Lemma 4.7 (p. 20).

Bears on

• Problem 1196: the $k$-almost primes form a primitive set lying in $[2^k,\infty)$, and the proposition gives its sum $\sum1/(n\log n)$ as $1+O(k/2^{k/4})$, so the sum tends to $1$. The sharper Theorem 1.2 also gives the sign of the error. The paper does not pose or answer the problem.