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For a given $N$, is there an approach to find the maximum $\frac{\phi(i)}i$ ($2\le i\le N$)?

Like for $n=2$, $\frac{\phi(2)}2=\frac12$ is maximum

for $n=3$, $\frac{\phi(3)}3=\frac23$ is maximum

for $n=4$, $\frac{\phi(3)}3=\frac23$ is maximum

From wikipedia's definition : $\frac{\phi(n)}n = \prod_{p|n}\left (1 - \frac1p\right)$.

So, I guess $p$ should be very big for this approach. Still, I can't find a generalised approach. Please help.

mohan
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2 Answers2

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Let $n\ge 2$ be given and $p$ the biggest prime $\le p$. Then $\frac{\phi(p)}p=1-\frac1p$. If $2\le i\le n$, then $i$ has $m\ge1$ prime factors $q_k\le p$ ($1\le k\le m$), hence $$\frac{\phi(i)}i=\prod_{k=1}^m\left(1-\frac1{q_k}\right)\le \left(1-\frac1p\right)^m\le 1-\frac1p.$$ Thus $1-\frac1p$ is indeed the maximum value: If $n\ge2$, then $$\max\left\{\frac{\phi(i)}i\biggm| 2\le i\le n\right\} =1-\frac1{\max\{p\mid p\le n, p\text{ prime}\}} $$

Hagen von Eitzen
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  • So, basically we will get the maximum value everytime for the biggest prime (just less or equal to n)? – mohan May 04 '13 at 10:54
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Yes this ratio will be maximum for the largest prime less than equal to N. Now the question is, how would you compute such prime when N is very large.

Achint
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