Superior Highly Composite Numbers and the Explicit Upper Bound of Generalized Divisor Functions
Lee-Peng Teo
TL;DR
This work establishes that the generalized divisor function $d_k(n)$ induces a bounded, maximized growth rate captured by $f_k(n)=\frac{\log d_k(n) \log\log n}{\log k \log n}$, with $\limsup_{n\to\infty} f_k(n)=\lambda(k)$ attained at a specific integer $N_{\max}(k)$. The authors extend Ramanujan’s and Nicolas’ framework to general $k$ by rigorously characterizing maximizers as superior $k$-highly composite numbers and deriving an explicit construction for the associated numbers $N_{k,\varepsilon}$ via $N_{k,\varepsilon}=\prod_{p\le k^{1/\varepsilon}} p^{m(p,k,\varepsilon)}$, where $m(p,k,\varepsilon)=\left\lfloor \frac{k-1}{p^{\varepsilon}-1}\right\rfloor$. They provide computable bounds on the admissible $\varepsilon$-range, prove the limit superior equals 1 in a general setting, and develop a finite, practical algorithm to determine $\lambda(k)$ and $N_{\max}(k)$ for each $k$, with explicit results up to $k\le100$ and comprehensive data for larger $k$. The numerical algorithm confirms $N_{\max}(k)$ stays within explicitly constructed candidate sets, and the computed $\lambda(k)$ values illuminate the behavior of generalized divisor growth. These results yield explicit upper bounds on $\log d_k(n)$ in terms of $\log n$, with potential applications in analytic number theory and divisor-function estimates.
Abstract
For $k\geq 2$, we give a detailed exposition of the superior $k$-highly composite numbers. We then consider the function \[f_k(n)=\frac{\log d_k(n)\log\log n}{\log k\log n},\quad n\geq 3\] which has a maximum value $λ(k)$ at a superior $k$-highly composite number. We develop an efficient algorithm to compute $λ(k)$ and the positive integer $N_{\max}(k)$ where $f_k$ achieves the value $λ(k)$. The results for $2\leq k\leq 100$ are tabled.