Inequalities involving the primorial counting function

Let φ(n) denote the Euler totient function. In this paper, we first establish a new upper bound for n/φ(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 σ(n). Furthermore, we give some lower bounds for Nₖ/φ(Nₖ) as well as for σ(Nₖ)/Nₖ, where Nₖ denotes the kth primorial.