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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2411.11065 |
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| _version_ | 1866929594849820672 |
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| author | Halupczok, Karin Ohst, Marvin |
| author_facet | Halupczok, Karin Ohst, Marvin |
| contents | We prove that for all constants $a\in\N$, $b\in\Z$, $c,d\in\R$, $c\neq 0$, the fractions $ϕ(an+b)/(cn+d)$ lie dense in the interval $]0,D]$ (respectively $[D,0[$ if $c<0$), where $D=aϕ(\gcd(a,b))/(c\gcd(a,b))$. This interval is the largest possible, since it may happen that isolated fractions lie outside of the interval: we prove a complete determination of the case where this happens, which yields an algorithm that calculates the amount of $n$ such that $\rad(an+b)|g$ for coprime $a,b$ and any $g$. Furthermore, this leads to an interesting open question which is a generalization of a famous problem raised by V.~Arnold. For the fractions $ϕ(an+b)/ϕ(cn+d)$ with constants $a,c\in\N,b,d\in\Z$, we prove that they lie dense in $]0,\infty[$ exactly if $ad\neq bc$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11065 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Density properties of fractions with Euler's totient function Halupczok, Karin Ohst, Marvin Number Theory We prove that for all constants $a\in\N$, $b\in\Z$, $c,d\in\R$, $c\neq 0$, the fractions $ϕ(an+b)/(cn+d)$ lie dense in the interval $]0,D]$ (respectively $[D,0[$ if $c<0$), where $D=aϕ(\gcd(a,b))/(c\gcd(a,b))$. This interval is the largest possible, since it may happen that isolated fractions lie outside of the interval: we prove a complete determination of the case where this happens, which yields an algorithm that calculates the amount of $n$ such that $\rad(an+b)|g$ for coprime $a,b$ and any $g$. Furthermore, this leads to an interesting open question which is a generalization of a famous problem raised by V.~Arnold. For the fractions $ϕ(an+b)/ϕ(cn+d)$ with constants $a,c\in\N,b,d\in\Z$, we prove that they lie dense in $]0,\infty[$ exactly if $ad\neq bc$. |
| title | Density properties of fractions with Euler's totient function |
| topic | Number Theory |
| url | https://arxiv.org/abs/2411.11065 |