The Vandermonde Determinant of the Divisors of an Integer

TL;DR

This work studies the Vandermonde determinant of the divisors of an integer, defining $V(n)=\det\mathcal{V}(d_1,\dots,d_{\tau(n)})=\prod_{1\le i<j\le \tau(n)}(d_j-d_i)$ where $d_1<\dots<d_{\tau(n)}=n$ are the divisors of $n$. The author derives sharp asymptotic bounds for $\log V(n)$ in terms of $\tau(n)$, $\log n$, and the quadratic divisor-function $\Omega_2(n)=\sum_{p^\alpha\|n}\alpha^2$, namely $\frac{\tau(n)^2}{4}(\log n)\left(1+O\left(\frac{1}{\log n}+\frac{1}{\sqrt{\Omega_2(n)}}\right)\right) \le \log V(n) \le \frac{3\tau(n)^2}{8}(\log n)$ for $n\ge 2$. The paper develops preliminary bounds for the related sum $S(n)=\sum_{i=1}^{\tau(n)}(i-1)\log d_i$, leveraging the identity $\sum_{d|n}\log d=\frac{\tau(n)}{2}\log n$, and uses symmetry of divisor pairs to obtain both lower and upper bounds; a key corollary provides an exact expression for $\sum_{d|n}(\log d-\tfrac{1}{2}\log n)^2$ in terms of the prime-power structure of $n$, from which the range of limit points of $\frac{S(n)}{\tau(n)^2\log n}$ is shown to be $[\tfrac{1}{4},\tfrac{3}{8}]$. Finally, the proof of the main theorem ties these estimates together, showing $\log V(n)$ is well approximated by $S(n)$ up to an error controlled by $\Omega_2(n)$, yielding the stated asymptotic bounds for $\log V(n)$. The results provide a precise growth description for the Vandermonde determinant of divisors and connect divisor-log distributions with arithmetic structure via $\Omega_2(n)$ and von Mangoldt-based expansions.

Abstract

Let $1=d_{1}<d_{2}< \cdots < d_{τ(n)}=n$ denote the ordered sequence of the positive divisors of an integer $n$. We are interested in estimating the arithmetic function $$ V(n) := \prod_{1 \le i < j \le τ(n)}(d_{j}-d_{i}) \quad (n \ge 1). $$

Theorems & Definitions (10)

• Theorem 1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Corollary 1
• proof
• Remark 1