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Source. Conjecture 1, p. 1, 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). $\mathcal D_n$ is the set of the $\tau(n)$ divisors of $n$, and

$$D_n(X,Y)=\lvert\{d\in\mathcal D_n: X\le d\le X+Y\}\rvert ,$$

the number of divisors of $n$ in the closed interval $[X,X+Y]$.

Conjecture 1 (p. 1, quoted). "Let $\epsilon>0$ be fixed. There exists a constant $k_\epsilon$ such that, for each integer $n\ge1$, we have $D_n(n^{1/2},n^{1/2-\epsilon})\le k_\epsilon$."

The paper presents it as suggested in the literature and cites Erdős and Rosenfeld (Acta Arith. 79 (1997)) and two papers of T. H. Chan (Acta Arith. 163 (2014); Int. J. Number Theory 11 (2015)). For $\epsilon<1/2$ it is the case $\theta=1/2$ of the paper's own Conjecture 2 (p. 1), which asks, for each fixed $0<\theta<1$ and $0<\epsilon<\theta$, for a constant $k_\epsilon(\theta)$ with $D_n(n^\theta,n^{\theta-\epsilon})\le k_\epsilon(\theta)$ for each integer $n\ge1$.

What the paper proves about it. Proposition 1 (p. 4) at $\theta=1/2$, with the uniformity in $n$ that its proof gives, yields the cases $1/4<\epsilon<1/2$ (Proposition 1; a specialization by this page, not a claim of the paper). For $\epsilon\ge1/2$ the window has length at most $1$ and holds at most two divisors. Theorem 2 (p. 2) at $\theta=1/2$ shows that constants as in Conjecture 2 must satisfy $k_\epsilon(1/2)\gg\sqrt\epsilon\,2^{1/\epsilon}$ (Theorem 2). Section 6 (pp. 12-14) discusses a method for windows near $\sqrt n$ without proving the conjecture. The case $0<\epsilon\le1/4$ is left open.

Read depth. Claims checked: the statement and its setting were read on the printed page. Not independently reviewed.

Bears on

• Problem 886: the problem asks whether, for each $\epsilon>0$, every large $n$ has $O_\epsilon(1)$ divisors in the open interval $(n^{1/2},n^{1/2}+n^{1/2-\epsilon})$. That count and $D_n(n^{1/2},n^{1/2-\epsilon})$ differ by at most $2$, and each of the finitely many small $n$ has finitely many divisors, so Conjecture 1 and a yes answer to Problem 886 are equivalent (an observation of this page, not of the paper). The paper does not prove the conjecture.
• Problem 887: the problem asks for an absolute $K$ such that, for every $C>0$, every large $n$ has at most $K$ divisors in $(n^{1/2},n^{1/2}+Cn^{1/4})$. For any one fixed $\epsilon$ with $0<\epsilon<1/4$, $Cn^{1/4}\le n^{1/2-\epsilon}$ once $n$ is large, so Conjecture 1 for that $\epsilon$ would give $K=k_\epsilon$ (an observation of this page, not of the paper). The conjecture is unproved in that range.