Inequalities involving the primorial counting function
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913380187504640 |
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| author | Axler, Christian |
| author_facet | Axler, Christian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_04018 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Inequalities involving the primorial counting function Axler, Christian Number Theory 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. |
| title | Inequalities involving the primorial counting function |
| topic | Number Theory |
| url | https://arxiv.org/abs/2406.04018 |