Inequalities involving the primorial counting function

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