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Statement

Setting (pp. 243, 250). $H(x,y,z)$ is the number of integers $n<x$ having at least one divisor $d$ with $y\le d<z$. In §4 the positive real $\eta$ is defined by $z=(1+\eta)y$, and $A(v)=v\log v-v+1$ on $\mathbb R_+$.

Theorem 2 (p. 250). Suppose $x$, $y$ and $z$ tend to infinity so that

$$0<\eta\le1,\qquad \eta y\to\infty,\qquad z\le\sqrt x.$$

Then:

(i) If there is a function $\xi(y)\to\infty$ with

$$\eta(\log y)^{\log4-1}\exp\bigl\{\xi(y)\sqrt{\log\log y}\bigr\}=o(1), \qquad(*)$$

then $H(x,y,z)=(1+o(1))\eta x$.

(ii) If $\gamma:=(\log1/\eta)(\log\log y)^{-1}\le\log4-1+o(1)$, then

$$H(x,y,z)=x(\log y)^{-A((1+\gamma)/\log2)+o(1)}.$$

Remark (p. 250). $A(1/\log2)=\delta$, so part (ii) with $\gamma=0$ is a slightly weakened form of Theorem 1 in the case $z=2y$.

Proof pointer

Pp. 250--253. Part (i): the upper bound comes from the mean $\sum_{n<x}\rho(n)=(1+o(1))\eta x$ of the number $\rho(n)$ of divisors of $n$ in $[y,z)$. For the lower bound, the case $\eta\log y=o(1)$ follows from a second-moment bound and Cauchy--Schwarz; otherwise the paper restricts to integers with few prime factors below $y$ and bounds the contribution of integers with two or more divisors in $[y,z)$, using Lemma 1 and the estimate (4) (p. 252). Part (ii): the paper says the method of §§6--7 applies, simplified because $y^u<2$ for $\eta\ne1$, and omits the details (p. 253).

Read depth

Claims checked: the definitions, the theorem and the remark were read clause by clause on the page image of p. 250. The proof of (i) was read for structure only; the proof of (ii) is omitted in the paper. Nothing here is independently reviewed.

Dependencies

Theorem 1 (method of §§6--7, for part (ii)); the paper's Lemma 1.

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: the theorem concerns intervals $[y,(1+\eta)y)$ with $0<\eta\le1$; at $\eta=1$, the problem's interval length, part (ii) with $\gamma=0$ gives $H(x,y,2y)=x(\log y)^{-\delta+o(1)}$, which the paper calls a slightly weakened form of Theorem 1. It says nothing about integers with exactly one divisor in the interval.