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arXiv · 2406.04018

Inequalities involving the primorial counting function

Christian Axler
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Abstract

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 $N_k/φ(N_k)$ as well as for $σ(N_k)/N_k$, where $N_k$ denotes the $k$th primorial.

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Christian Axler. 2024-06-06. Inequalities involving the primorial counting function. https://arxiv.org/abs/2406.04018

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