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That is, is it true that the bound $$\phi(mn)\leq m\phi(n)$$ holds for all pairs of natural numbers $m$ and $n$?

It is true on average, in the sense that $$\sum_{mn=k}m\phi(mn)\leq\sum_{mn=k}mn\phi(n),$$ and it holds for every pair I have computed. If it does not hold, what is the smallest counter example?

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Another way of seeing this: $\phi(mn)$ is the number of integers in $[1,mn]$ coprime to $mn$, while $m\phi(n)$ is the number of integers in $[1,mn]$ coprime to $n$. –  Anonymous Feb 5 '13 at 1:50
Thanks, Anonymous. You can also use Euler's product to get it directly, which provides the exact ratio. This I noticed following Quid's answer below. I'd like to point out that my interest was in the not-so-obvious corollary, that $$\sup_{d|n}\frac{d}{\phi(d)}=\frac{n}{\phi(n).$$ –  Kevin Smith Feb 6 '13 at 10:13

1 Answer 1

up vote 2 down vote accepted

First note that, since $\phi$ is a multiplicative arithmetic function, i.e. $\phi(ab)=\phi(a)\phi(b)$ for coprime $a,b$, and since $\phi(a) \le a$ for every $a$, it suffices to consider the problem for prime powers.

And $$\phi(p^k p^t) = p^{k+t-1}(p-1) = p^k (p^{t-1}(p-1))= p^k\phi(p^t).$$

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Thanks Quid, I had hoped there was an easy way. –  Kevin Smith Feb 2 '13 at 14:52
@Kevin Smith: you are welcome! –  quid Feb 2 '13 at 15:37

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