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

Abstract

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

50 pages, 20 tables, 1 figure. Accepted by the Ramanujan Journal