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

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 $\lambda(k)$ 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 $2\leq k\leq 100$ are tabled.