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Does any of you know if the inequality

$\displaystyle \frac{\phi(\sigma(n))}{n} < (\log \log \log n)^{-1/2}$

is true for all $n$ sufficiently large?

I remember reading something to that effect sometime ago, but a detailed statement of the result eludes me now and that's the reason that you find me asking it here.

I thank you all for your replies.

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1 Answer

I'm guessing that $\sigma(n)$ is supposed to be the divisor function and $\phi(n)$ is the Euler-$\phi$ function. In which case, assuming the conjecture that there are infinitely many prime pairs $(p,q)$ with $p + 1 = 2q$, then $$\phi(\sigma(p))/p = \phi(p+1)/p = (q-1)/p \sim \frac{1}{2}.$$ If you want to disprove the bound unconditionally, an easy sieving argument shows that there are infinitely many primes $p$ such that $p+1$ is a product of at most (say) $100$ primes, and one obtains a similar estimate with $1/2$ replaced by the product $$\prod_{n=1}^{100} \left(1 - \frac{1}{p_n} \right)$$ over the first $100$ primes. Perhaps you were remembering a statement about almost all $n$? The wrong sign in the exponent?

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I think "for almost all n" is what I had in mind. Thanks. –  J. H. S. Oct 23 '10 at 18:45
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