Inequalities involving the primorial counting function

Abstract

Let $\phi (n)$ denote the Euler totient function. In this paper, we first establish a new upper bound for $n/\phi (n)$ involving $K(n)$, the function that counts the number of primorials not exceeding $n$. In particular, this leads to an answer to a question raised by Aoudjit, Berkane, and Dusart concerning an upper bound for the sum-of-divisors function $\sigma (n)$. Furthermore, we give some lower bounds for $N_k/\phi (N_k)$ as well as for $\sigma (N_k)/N_k$, where $N_k$ denotes the $k$th primorial.