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Statement

Setting (p. 339, §5). For $i\geq1$ let $e_i$ be the density of the set of all integers having a divisor $\geq2^i$ and $<2^{i+1}$. (This set is the set of multiples of the integers in $[2^i,2^{i+1})$; it is periodic, so its density exists.)

Theorem 1 (p. 339, quoted).

$$e_1+e_2+\cdots+e_l=o(l).$$

The limit is $l\to\infty$. The paper adds (p. 340): "As every $e_i>0$ we conclude that $e_i$ is small for almost all $i$." In particular $\liminf_{i\to\infty}e_i=0$, so for every $\eta>0$ there are arbitrarily large $i$ with $e_i<\eta$; this is the form used in §7 (p. 340).

The introduction (p. 336) frames the theorem as the density form of a "probability argument": by the Hardy--Ramanujan theorem on the normal order of $d(n)$, for a positive integer $a$ almost all $n$ have divisors in only $o(\log n)$ of the intervals from $a^i$ to $a^{i+1}$.

Source. A. S. Besicovitch, "On the density of certain sequences of integers," Mathematische Annalen 110 (1935), 336--341, https://doi.org/10.1007/BF01448032: the notation of §1 on pp. 336--337, Lemmas 1--3 on pp. 337--338, the estimate (5) on p. 339, Theorem 1 on p. 339 and its proof on pp. 339--340. The edition read is identified on the source card.

Read depth. Claims checked: the definition of $e_i$, the statement and the remark after the proof were read on the printed pages. The proof was read but not checked step by step. Nothing here is independently reviewed.

Proof pointer

Pages 337--340. Lemma 1 (p. 337): a set $F$ of density zero has $\sum_{\nu\in F,\,\nu\leq n}1/\nu=o(\log n)$. Lemma 2 (p. 337): the product $\prod_{p<2n}(1-p^{-1})^{-1}$ lies between two constant multiples of $\log n$. Lemma 3 (p. 338, its proof credited in a footnote to H. Davenport): with $H(2n)$ the integers having no prime factor above $2n$, the sum of $1/\nu$ over $\nu\in H(2n)$ with $\nu\geq n^k$ is less than $(B/k)\log n$ for all $n$ and $k$, with a constant $B>0$. The paper calls a number $\nu$ with $d(\nu)>(\log\nu)^{\log2.1}$ highly composed (§4, p. 338); these have density zero by Hardy--Ramanujan, so removing them from $H(2n)$ loses little of the harmonic sum up to $n^{k_0}$, which is (5) (p. 339).

For the theorem, take $n=2^l$ and a factorial period $N$, and let $G$ be the integers up to $N$ whose $(<2n)$-smooth part is a non-highly-composed number at most $n^{k_0}$; by (5) these are all but $\varepsilon N$ of the integers up to $N$, (6). Counting the divisors below $2n$ of members of $G$ in two ways gives a lower bound $(e_1+\cdots+e_l-l\varepsilon)N$, (7), and an upper bound $(\log n^{k_0})^{\log2.1}N$, (8), since each such number has at most $d(\nu)$ divisors below $2n$. As $\log 2.1<1$, the upper bound is $o(l)N$, and the theorem follows.

Dependencies

The Hardy--Ramanujan theorem on the normal order of $d(n)$, cited on p. 336 to the Collected Papers of Srinivasa Ramanujan, pp. 261--275; Lemmas 1--3 of the same paper.

Bears on

• Problem 446: the problem's $\delta(n)$ is the density of the integers divisible by some integer in $(n,2n)$, so $\delta(2^i)\leq e_i$ and Theorem 1 gives $\liminf_{n\to\infty}\delta(n)=0$ (an observation of this page). It gives neither $\delta(n)\to0$ nor a growth rate, which the problem asks for.
• Problem 25: Theorem 1 is the input to the §7 construction of a set of multiples without natural density; see the construction page, which states the relation to the problem.