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Statement

Setting (p. 243). $H(x,y,z)$ is the number of integers $n<x$ having at least one divisor $d$ with $y\le d<z$.

Theorem 3 (p. 253). For $1<y\le z\le x$,

$$H(x,y,z)=x\Bigl(1+O\Bigl(\frac{\log y}{\log z}\Bigr)\Bigr).$$

Moreover, for every $\epsilon$ with $0<\epsilon<1$, under the conditions $y^\epsilon z<x$ and $y\ge y_0(\epsilon)$,

$$x-H(x,y,z)\gg\epsilon x\,\frac{\log y}{\log z}.$$

The implied constants are absolute (p. 246, §2).

Proof pointer

P. 253. The first assertion: an integer with no prime factor in $[y,z)$ is the only kind that can fail to be counted, and Lemma 7 bounds those by $x\exp\{-\sum_{y\le p<z}1/p\}\ll x\log y/\log z$. The second: integers $mn$ with $m<c_9y^\epsilon$ and $P^-(n)\ge z$ have no divisor in $[y,z)$, and Lemma 5 counts them.

Read depth

Claims checked: the theorem was read clause by clause on the page image of p. 253 and its short proof followed. Nothing here is independently reviewed.

Dependencies

The paper's Lemmas 5 and 7.

Source. G. Tenenbaum, Sur la probabilité qu'un entier possède un diviseur dans un intervalle donné, Compositio Math. 51 (1984), no. 2, 243--263; the edition read is named on the source card.

Bears on

None. The theorem is informative when $\log y=o(\log z)$, and the paper does not relate it to an Erdős problem.