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Statement

Setting (p. 340, §6). Take a number $\alpha$ with $\log2<\alpha<1$ and "a sequence of positive integers"

$$n_1,\quad n_2=n_1^{1+\log^{-\alpha}n_1},\quad\ldots,\quad n_{i+1}=n_i^{1+\log^{-\alpha}n_i},\quad\ldots$$

(as printed, with $\log^{-\alpha}n_i=(\log n_i)^{-\alpha}$; the paper does not say how the right sides are made integers). Let $m_i$ be the density of the set of integers with a divisor $\geq n_i$ and $<n_{i+1}$.

Theorem 2 (p. 340, quoted).

$$m_1+m_2+\cdots+m_l=o(l).$$

The limit is $l\to\infty$. The paper calls this theorem "much stronger than Theorem 1" and gives no separate proof: "the proof is practically identical with the proof of Theorem 1" (p. 340).

Source. A. S. Besicovitch, "On the density of certain sequences of integers," Mathematische Annalen 110 (1935), 336--341, https://doi.org/10.1007/BF01448032: Theorem 2 and the remark after it on p. 340. The edition read is identified on the source card.

Read depth. Claims checked: the hypotheses and the statement were read on the printed page. The paper prints no proof, and none was checked here. Nothing here is independently reviewed.

Proof pointer

None in the paper beyond the remark that the proof of Theorem 1 (pp. 339--340) carries over.

Dependencies

The proof of Theorem 1 of the same paper, by the paper's remark.