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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 1 (pp. 243--244). Put $\delta:=1-\log(e\log2)/\log2=0.08607\ldots$. Under the hypothesis

$$1<2y\le z\le\min\bigl(y^{3/2},x^{1/2}\bigr)\qquad(1)$$

and with $u$ defined by $z=y^{1+u}$,

$$x\,u^\delta L_1(1/u)<H(x,y,z)<x\,u^\delta L_2(1/u),\qquad(2)$$

where $L_1$ and $L_2$ are slowly varying functions tending to $0$ at infinity, one possible choice being

$$L_1(v)=\exp\Bigl(-c_1\sqrt{\log v\,\log\log2v}\Bigr),\qquad L_2(v)=c_2(\log v)^{-1/2}\log\log2v,$$

with $c_1$ and $c_2$ positive constants. Moreover, when $z=O(y)$ the factor $\log\log2v$ may be omitted from $L_2(v)$.

Remarks after the theorem (pp. 244--245). In condition (1), $2y$ may be replaced by $(1+\eta)y$ for a fixed real $\eta>0$, the constants $c_1$ and $c_2$ then depending on $\eta$. The paper also says, without proof, that the condition $z\le x^{1/2}$ can be relaxed using the symmetry of the divisors of $n$ about $n^{1/2}$, and that the exponent $3/2$ can be replaced by any constant greater than $1$ after suitably changing $L_1$ and $L_2$ for small $v$. The paper states that the theorem strictly contains the earlier results on the four special cases it lists (p. 243).

Proof pointer

The upper bound is §6 (pp. 254--257): each counted integer is written $ab$ with $P^+(a)\le y^u<P^-(b)$, Lemma 3 lets $a$ be taken small, and the integers are split into four classes by $n(y^u)$ and by the number of prime factors in $[y^u,y)$, the classes bounded in (5), (7), (9) and (10); the case $z=O(y)$ follows from the case $z\le cy$, $c<2$, where two of the classes are empty. The lower bound is §7 (pp. 257--263): a Cauchy--Schwarz inequality (11) over a set $S$ of integers with a controlled number of prime factors in each range, with the first and second moments of a weighted divisor count bounded in Lemmas 10 and 11.

Read depth

Claims checked: the theorem, condition (1), the choices of $L_1$, $L_2$ and the remarks on pp. 244--245 were read clause by clause on the page images of the print. The proofs of §§6--7 were read for structure only. Nothing here is independently reviewed.

Dependencies

The paper's Lemmas 1--11 (§3 and §7); Lemma 1 is a weakened form of a theorem of Halberstam and Richert.

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

• Problem 446: taking $z=2y$ in (2), so that $u=\log2/\log y$, bounds $H(x,y,2y)/x$ above and below by $u^\delta$ times the slowly varying factors $L_2(1/u)$ and $L_1(1/u)$, for $y\ge4$ and $x\ge4y^2$, where (1) holds. This gives the growth rate of the problem's density up to those factors, not its order of magnitude; the paper's interval is $[y,2y)$, the problem's is $(n,2n)$.