On integral boxes of minimal surface

TL;DR

The paper addresses the problem of minimizing the surface of a k-dimensional box with integral edge lengths and fixed volume by identifying the optimal edge lengths $\varrho_h(n)$ solving the constraint $\prod_{h=1}^k \varrho_h(n)=n$. It derives precise mean-value asymptotics for $\sum_{n\le x} \varrho_j(n)$: for $2\le j\le k$ the sum behaves like $\frac{\gamma_j^{2\gamma_j}}{(\gamma_j+1)!} \zeta(1+1/\gamma_j) \frac{x^{1+1/\gamma_j}}{(\log x)^{\gamma_j}}$ with $\gamma_j=k+1-j$, while $\sum_{n\le x} \varrho_1(n)$ has a distinct growth rate $\asymp_k \frac{x^{1+1/k}}{(\log x)^{\delta_k}(\log_2 x)^{3/2}}$ and $\delta_k=Q((k-1)/\log k)$. The authors develop the analysis via a dyadic partitioning strategy, deep results on mean-values of divisor-like structures, and a key lemma that constrains optimality to specific prime-factor configurations, enabling explicit asymptotics through a detailed evaluation of auxiliary sums such as $T_j(y)$. These findings extend the known 2D results to higher dimensions and illuminate how minimal-surface box configurations correspond to arithmetic-structure constraints. The work connects geometric optimization with exact asymptotics in arithmetic function theory, offering precise constants and illuminating the role of primes in the distribution of optimal edge lengths.

Abstract

Generalising the two-dimensional case, we provide estimates for the mean-values of the lengths of the edges of an integral box with given volume and minimal surface.

Theorems & Definitions (10)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 3.1
• proof
• Lemma 4.1
• proof
• Lemma 4.2
• proof