Divisors of an Integer in a Short Interval

TL;DR

This work studies how many divisors of an integer $n$ lie in a short interval $[X,X+Y]$ via the function $D_n(X,Y)=|\{d\in\mathcal{D}_n:\ X\le d\le X+Y\}|$, focusing on $Y\le X$ and $D_n(n^{\theta},n^{\eta})$. The author proves a general upper bound $D_n(n^{\theta},n^{\eta}) \ll \tau(n)^{1-\xi(\theta,\eta)} \dfrac{V(n)\log\tau(n)}{\theta(1-\theta)}$ with a piecewise exponent $\xi(\theta,\eta)$, revealing how divisor structure controls short-interval concentration; a relaxed variant and a detailed parameter analysis are provided. A conditional lower bound is established under a conjecture on short-interval divisor counts, giving a quantified growth rate for $k_{\varepsilon}(\theta)$. The methods blend divisor combinatorics, sieve-type bounds, and congruence-class arguments, and the discussion connects to the isolated nature of large divisors and a large-sieve framework, yielding a cohesive view of divisor distribution in short ranges. These results deepen understanding of the interplay between divisor counts, interval size, and the arithmetic structure of $n$, with implications for related conjectures and further sieve-based approaches.

Abstract

Let $\mathcal{D}_{n} \subset \mathbb{N}$ denote the set of the $τ(n)$ divisors of $n$. We study the function $$ D_{n}(X,Y):=|\{d \in \mathcal{D}_{n}:\ X \le d \le X+Y\}| $$ for $Y \le X$.

Theorems & Definitions (20)

• Conjecture 1
• Conjecture 2
• Theorem 1
• Theorem 2
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof