Superior Highly Composite Numbers and the Explicit Upper Bound of Generalized Divisor Functions

For $k\ge 2$, we give a detailed exposition of the superior $k$-highly composite numbers. We then consider the function $$f_k(n)=\frac{\log d_k(n)\log \log n}{\log k\log n},n\ge 3$$ which has a maximum value $\lambda (k)$ at a superior $k$-highly composite number. We develop an efficient algorithm to compute $\lambda (k)$ and the positive integer $N_{\max }(k)$ where $f_k$ achieves the value $\lambda (k)$. The results for $2\le k\le 100$ are tabled.