Inequalities involving the primorial counting function

2024/06/06 by Christian Axler, Axler, Christian
Mathematics · #FOS: Mathematics #Functional Equations Stability Results #Mathematical Inequalities and Applications #Number Theory (math.NT) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2406.04018

openalex publication_date 2024/06/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 kth primorial.

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