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Source. Proposition 1, p. 4, of Patrick Letendre, Divisors of an Integer in a Short Interval, arXiv preprint arXiv:2503.12146v1 (15 March 2025), the version named on the source card.

Statement

Setting (p. 1). $D_n(X,Y)$ is the number of divisors $d$ of $n$ with $X\le d\le X+Y$.

Proposition 1 (p. 4, quoted). "Let $n\ge1$ be a fixed integer, and let $0<\theta<1$ and $0<\epsilon<\theta^2$ be fixed real numbers. Then

$$D_n(n^\theta,n^{\theta^2-\epsilon})\ll\frac{\theta(1-\theta)}{\epsilon}+\frac{1}{\theta(1-\theta)}."$$

The right side does not involve $n$. The printed statement fixes $n$ and does not name the dependence of the implied constant. The proof (p. 4) ends with the alternative $k\le4/(\theta(1-\theta))$ or $k\le3\,\theta(1-\theta)/\epsilon$ for the number $k\ge2$ of divisors in the window, with no dependence on $n$, so the bound holds uniformly in $n$ (a reading of the proof by this page).

The paper states it in Section 4 as one of three statements used for Theorem 1. On p. 2 it refers to Proposition 1 when it says that a relaxed version of Conjecture 2 would ask for a larger region in which $\xi(\theta,\eta)=1$; in Theorem 1's notation the proposition covers windows of length $n^\eta$ with $\eta<\theta^2$, inside the region $\eta\le\theta^2$ where $\xi(\theta,\eta)=1$.

Read depth. Claims checked: the statement was read clause by clause on the printed page, and the short proof was read. Not independently reviewed.

Proof pointer

Page 4. Take divisors $d_1,\ldots,d_k$ of $n$ in the window. Their least common multiple is at most $n$, each pairwise greatest common divisor is at most the gap $\lvert d_i-d_j\rvert\le n^{\theta^2-\epsilon}$, and each $d_i\ge n^\theta$. Lemma 1 (p. 2) with $t=\theta k+\zeta$, $0\le\zeta<1$, then gives, on comparing exponents of $n$, $\epsilon k(k-1)\le(\theta-\theta^2)k+\zeta(\zeta+1)$, which forces one of the two bounds on $k$ above.

Dependencies

Lemma 1 (p. 2): for positive integers $d_1,\ldots,d_k$ and every integer $t$, $[d_1,\ldots,d_k]^{t(t+1)/2}\prod_{1\le i<j\le k}(d_i,d_j)\ge\prod_{1\le i\le k}d_i^t$, where $[\cdot]$ is the least common multiple and $(\cdot,\cdot)$ the greatest common divisor. The paper takes it from H. Cohen, Diviseurs appartenant à une même classe résiduelle, Seminar on number theory 1982-83, Université de Bordeaux I, Exp. No. 16, Corollaire 1.4.

Bears on

• Problem 886: at $\theta=1/2$ the proposition gives, for each fixed $0<\delta<1/4$, at most $O(1/\delta)$ divisors of $n$ in $[n^{1/2},n^{1/2}+n^{1/4-\delta}]$, for every $n$. This answers the problem's question for each $\epsilon=1/4+\delta$ with $1/4<\epsilon<1/2$; for $\epsilon\ge1/2$ the window has length at most $1$. The problem page records that Erdős and Rosenfeld's bound already answers the question for every $\epsilon\ge1/4$. The range $0<\epsilon\le1/4$ would need windows of length $n^\eta$ with $\eta\ge\theta^2$, outside the proposition's hypothesis.
• Problem 887: the problem's windows have length $Cn^{1/4}$, longer than the $n^{1/4-\delta}$ the proposition allows at $\theta=1/2$, so it settles no instance.
• Problem 873: the paper does not mention this problem. A note posted in the problem's thread (claim page) combines this proposition with packing lemmas of its own in an argument that it says answers the question for every exponent above $1/4$; that reduction is the note's, not the paper's, and is not checked here.