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Can someone help me to evaluate the following sum. It appeared while calculating the number of elements of intersections of some sets: $$\sum_{d|n} \frac{d}{\phi(d)^2}.$$

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It looks to be a multiplicative function, and the result for prime powers is somewhat tractable. Have you already looked at it to that extent? –  hardmath Jul 7 '12 at 23:28
    
To extend on hardmath's finding I got indeed $f(p)=\frac{p^2-p+1}{(p-1)^2}$ (the 'distribution' obtained is very regular with a mean value of $\approx 3.39063$) –  Raymond Manzoni Jul 7 '12 at 23:34
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1 Answer 1

Since $\dfrac{n}{\phi(n)^2}$ is a multiplicative function, we know that $$ \varphi(n)=\sum_{d|n}\frac{d}{\phi(d)^2}\tag{1} $$ is also multiplicative. On the power of a prime, $p^j$ and $j>0$, $\phi(p^j)=p^j\frac{p-1}{p}$, so $$ \begin{align} \varphi(p^k) &=1+\sum_{j=1}^k\frac{p^2}{p^j(p-1)^2}\\ &=1+\frac{p}{(p-1)^2}\sum_{j=0}^{k-1}\frac{1}{p^j}\\ &=1+\frac{p^k-1}{p^{k-2}(p-1)^3}\tag{2} \end{align} $$ Thus, for $n$ whose prime factorization is $$ n=\prod_jp_j^{k_j}\tag{3} $$ we have $$ \varphi(n)=\prod_j\left(1+\frac{p_j^{k_j}-1}{p_j^{k_j-2}(p_j-1)^3}\right)\tag{4} $$ Estimation:

Note that for $k>0$, $$ 1+\frac1p\left(\frac{p}{p-1}\right)^2\le1+\frac{p^k-1}{p^{k-2}(p-1)^3}\lt1+\frac1p\left(\frac{p}{p-1}\right)^3\tag{5} $$ For larger primes, $(5)$ might help to approximate the term in $(4)$.

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+1 That's the expression I get. –  hardmath Jul 8 '12 at 0:00
    
An equivalent for $f(p^k)$ is $1 + \frac{p^2}{(p-1)^3} (1 - p^{-k})$, which makes it clearer what the limit is as $k$ tends to $\infty$. –  hardmath Jul 8 '12 at 0:18
    
@hardmath: thanks! I was just writing up an estimate along those lines. –  robjohn Jul 8 '12 at 0:32
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