Is $n_{\rightarrow\infty}\overline{\frac{n}{\phi (n)}}$ unbounded?

Question

Does a $\limsup_{n\to\infty}$ exist for $\overline{\dfrac{n}{\phi (n)}},$ where $\phi$ is the Euler totient function, and the overbar represents the mean.

Of course $\dfrac{n}{\phi (n)}$ is unbounded, as it grows like $e^{\gamma } \log (\log (x))+\dfrac{\text{c}}{\log (\log (x))}$ for some $c\approx 3,$ where $c\rightarrow 0$ as $n\rightarrow \infty,$ but I was wondering whether the set of (mostly, priomorial multiples) is dense enough to impose a limit on the function.

Numerical tests suggest $\limsup_{n\to\infty}\overline{\dfrac{n}{\phi (n)}}\approx 2,$ but I can't find any references to the subject. Is this realistic, or is the function unbounded?

By "mean" you mean $$\frac{1}{n}\sum_{k = 1}^n \frac{k}{\phi(k)},?$$ — Daniel Fischer, Nov 13 '15 at 14:39
@EricNaslund thanks for link to dup. - great answer there too (+1) :) — martin, Nov 13 '15 at 15:22

score 6 · Accepted Answer · answered Nov 13 '15 at 14:52

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We have that holds $$\sum_{k=1}^{n}\frac{k}{\phi\left(k\right)}=\frac{315\zeta\left(3\right)}{2\pi^{4}}n+O\left(\log\left(n\right)\right) $$ (see here for reference) then $$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{n}\frac{k}{\phi\left(k\right)}=\frac{315\zeta\left(3\right)}{2\pi^{4}}.$$

answered Nov 13 '15 at 14:52

Marco Cantarini

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that was quick! Thanks :) – martin Nov 13 '15 at 14:54
PDF link for your link above - thanks for ref. - looks like an interesting read :) – martin Nov 13 '15 at 14:57
That constant is close to 2 so good initial guess @martin. Its not a number I would have guessed :) – Ian Miller Nov 13 '15 at 14:57
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I expect the factor $\frac{315\zeta(3)}{2\pi^4}$ should be written [as in "comes from"] as $\frac{\zeta(2)\zeta(3)}{\zeta(6)}$. – Daniel Fischer Nov 13 '15 at 14:58
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@martin It's my pleasure. I'm glad that the reference is interesting for you ;) – Marco Cantarini Nov 13 '15 at 14:59
Elegant addition there @DanielFischer. – Ian Miller Nov 13 '15 at 15:01
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This is also proven here on Math.StackExchange: http://math.stackexchange.com/questions/84708/on-the-mean-value-of-a-multiplicative-function-prove-that-sum-limits-n-leq-x/84721#84721 – Eric Naslund Nov 13 '15 at 15:16

Is $n_{\rightarrow\infty}\overline{\frac{n}{\phi (n)}}$ unbounded?

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