Tenenbaum 1984 sur la probabilite qu un

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problem_p246: The open problem, attributed to Erdős, that closes the paper's introduction: it asks whether the density of integers with exactly one divisor in [y, 2y) is o(1) times the density of integers with at least one such divisor, as y tends to infinity.

theorem_1: Tenenbaum's theorem that, with z = y^{1+u} and delta = 0.08607..., the number H(x, y, z) of integers below x with a divisor in [y, z) lies between x u^delta L_1(1/u) and x u^delta L_2(1/u) for explicit slowly varying L_1, L_2 tending to 0, whenever 1 < 2y <= z <= min(y^{3/2}, x^{1/2}).

theorem_2: Tenenbaum's theorem on short intervals z = (1 + eta)y: as x, y, z tend to infinity with 0 < eta <= 1, eta y tending to infinity and z at most the square root of x, H(x, y, z) = (1 + o(1)) eta x under condition (*), and H(x, y, z) = x (log y)^{-A((1+gamma)/log 2)+o(1)} when gamma = log(1/eta)/log log y is at most log 4 - 1 + o(1).

theorem_3: Tenenbaum's theorem that H(x, y, z) = x(1 + O(log y/log z)) for 1 < y <= z <= x, and that x - H(x, y, z) is at least a constant times epsilon x log y/log z when 0 < epsilon < 1, y^epsilon z < x and y >= y_0(epsilon).

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Tenenbaum, G., Sur la probabilité qu'un entier possède un diviseur dans un intervalle donné. Compositio Math. 51 (1984), no. 2, 243-263. The copy read prints "© Foundation Compositio Mathematica, 1984, tous droits réservés." on its Numdam cover page (PDF p. 1) and "© 1984 Martinus Nijhoff Publishers, The Hague. Printed in The Netherlands" on the article's first page (PDF p. 2), every other right reserved.

Written in French, the paper studies $H(x,y,z)$, the number of integers $n<x$ with at least one divisor $d$ in $y\le d<z$ (p. 243). Theorem 1 (pp. 243--244) sets $\delta=1-\log(e\log2)/\log2=0.08607\ldots$ and, under $1<2y\le z\le\min(y^{3/2},x^{1/2})$, writing $z=y^{1+u}$, proves $xu^\delta L_1(1/u)<H(x,y,z)<xu^\delta L_2(1/u)$ with slowly varying functions $L_1$, $L_2$ tending to $0$, one possible choice being given explicitly in terms of positive constants $c_1$, $c_2$; the factor $\log\log2v$ in $L_2$ may be dropped when $z=O(y)$. The author says the theorem strictly contains all earlier results, which treated four special cases ($z=2y$ with $y$ fixed; $z=2y=\sqrt x$; $z=y^{1+u}$ with $u=o(1)$; $y$ and $z$ fixed powers of $x$). After the theorem the paper remarks that $2y$ may be replaced by $(1+\eta)y$ for a fixed $\eta>0$, with $c_1$, $c_2$ then depending on $\eta$ (p. 244).

Theorem 2 (§4, p. 250) treats $z=(1+\eta)y$ with $0<\eta\le1$, $\eta y\to\infty$ and $z\le\sqrt x$: under a smallness condition $(*)$ on $\eta$, $H(x,y,z)=(1+o(1))\eta x$; and when $\gamma=(\log1/\eta)/\log\log y\le\log4-1+o(1)$, $H(x,y,z)=x(\log y)^{-A((1+\gamma)/\log2)+o(1)}$ with $A(v)=v\log v-v+1$. Theorem 3 (§5, p. 253) gives $H(x,y,z)=x(1+O(\log y/\log z))$ for $1<y\le z\le x$, with a matching lower bound for $x-H(x,y,z)$. The upper bound of Theorem 1 is proved in §6 (pp. 254--257) and the lower bound in §7 (pp. 257--263). Among earlier work on the case $z=2y$ the paper cites Erdős's 1960 paper (its reference [5]). The introduction closes (p. 246) with an open problem attributed to Erdős: whether $\epsilon'(y)/\epsilon(y)=o(1)$, where $\epsilon(y)$ and $\epsilon'(y)$ are the densities of the integers with at least one and with exactly one divisor in $[y,2y)$.

Source: https://www.numdam.org/item/CM_1984__51_2_243_0/.

Read status: claims checked for Theorems 1, 2 and 3, the remarks on pp. 244--245 and 250, and the open problem on p. 246, read clause by clause on the page images of the print; the proof of Theorem 3 followed; the proofs of Theorems 1 and 2 read for structure only. Nothing here is independently reviewed.

Bears on. #446: Theorem 1 with $z=2y$ bounds $H(x,y,2y)/x$ above and below by $u^\delta$, $u=\log2/\log y$, times slowly varying factors, which 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 $(n,2n)$. The open problem on p. 246 is the problem's second question, posed for $[y,2y)$ and left open.

Results.

• Theorem 1 (pp. 243--244): under $1<2y\le z\le\min(y^{3/2},x^{1/2})$ and $z=y^{1+u}$, $xu^\delta L_1(1/u)<H(x,y,z)<xu^\delta L_2(1/u)$.
• Theorem 2 (p. 250): $H(x,y,(1+\eta)y)$ for $0<\eta\le1$, asymptotic to $\eta x$ under $(*)$, and equal to $x(\log y)^{-A((1+\gamma)/\log2)+o(1)}$ when $\gamma\le\log4-1+o(1)$.
• Theorem 3 (p. 253): $H(x,y,z)=x(1+O(\log y/\log z))$ for $1<y\le z\le x$, with a lower bound for $x-H(x,y,z)$.
• Open problem (p. 246): is $\epsilon'(y)/\epsilon(y)=o(1)$ as $y\to\infty$?

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