Inequalities involving the primorial counting function

TL;DR

This paper establishes new bounds relating the Euler totient function, primorial counting function, and sum-of-divisors function, providing insights into their inequalities and answering a previously posed question.

Contribution

It introduces a new upper bound for n/φ(n) involving the primorial counting function and addresses an open question about the sum-of-divisors function.

Findings

New upper bound for n/φ(n) involving K(n)

Answer to a question on the sum-of-divisors function σ(n)

Lower bounds for N_k/φ(N_k) and σ(N_k)/N_k

Abstract

Let $\varphi(n)$ denote the Euler totient function. In this paper, we first establish a new upper bound for $n/\varphi(n)$ 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 $N_k/\varphi(N_k)$ as well as for $\sigma(N_k)/N_k$, where $N_k$ denotes the $k$th primorial.