Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.23882 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913921909129216 |
|---|---|
| author | Gao, Pei Yang, Qiyu |
| author_facet | Gao, Pei Yang, Qiyu |
| contents | Let $ϕ(n)$ be the Euler totient function and $ϕ_k(n)$ its $k$-fold iterate. In this note, we improve the upper bound for the number of positive $n\leqslant x$ such that $ϕ_{k+1}(n)\geqslant cn$. Comparing with the upper bound which was obtained from Pollack's asymptotic formula of the summation of $ϕ_{k+1}(n)$ for $n\leqslant x$, we have successfully increased the denominator exponent of the main term of the upper bound from $k$ to $k+1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_23882 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An improved upper bound for the distribution of iterated Euler totient functions Gao, Pei Yang, Qiyu Number Theory 11N37 Let $ϕ(n)$ be the Euler totient function and $ϕ_k(n)$ its $k$-fold iterate. In this note, we improve the upper bound for the number of positive $n\leqslant x$ such that $ϕ_{k+1}(n)\geqslant cn$. Comparing with the upper bound which was obtained from Pollack's asymptotic formula of the summation of $ϕ_{k+1}(n)$ for $n\leqslant x$, we have successfully increased the denominator exponent of the main term of the upper bound from $k$ to $k+1$. |
| title | An improved upper bound for the distribution of iterated Euler totient functions |
| topic | Number Theory 11N37 |
| url | https://arxiv.org/abs/2506.23882 |