Finding the closed form $\phi(n) / n^x$

Question

$\sum_{1}^{\inf}{\phi(n)/n^x}$ Get an closed form expression for the value in terms of x.

Also, is there a closed form for the function $f(x) = \sum_{1}^{\inf}{\pi(n)/n^x}$. If you use PNT you can get some expressions in terms of li integrals but that's probably a dead end and since PNT is only an approximation its not the actual function.

Answer 1

Not a full answer, just an explaination on the first part. Let's look at

$$\left(\frac1{1^s}+\frac1{2^s}+\frac1{3^s}+\frac1{4^s}+\cdots\right)\left(\frac{\phi(1)}{1^s}+\frac{\phi(2)}{2^s}+\frac{\phi(3)}{3^s}+\frac{\phi(4)}{4^s}+\cdots\right)$$

how many times do we get a $\frac1{n^s}$? Well, for each $d\mid n$, we have the $k$ with $dk=n$, so that $\frac1{k^s}\frac{\phi(d)}{d^s}=\frac{\phi(d)}{n^s}$, so we get each $\frac1{n^s}$ exactly $\sum_{d\mid n}\phi(d)$ times, and, well, that's simply $n$. Thus, the above product is

\begin{align} &\ \ \ \ \ \ \left(\frac1{1^s}+\frac1{2^s}+\frac1{3^s}+\frac1{4^s}+\cdots\right)\left(\frac{\phi(1)}{1^s}+\frac{\phi(2)}{2^s}+\frac{\phi(3)}{3^s}+\frac{\phi(4)}{4^s}+\cdots\right)\\ &=\frac{1}{1^s}+\frac{2}{2^s}+\frac{3}{3^s}+\frac{4}{4^s}+\cdots\\ &=\frac{1}{1^{s-1}}+\frac1{2^{s-1}}+\frac1{3^{s-1}}+\frac1{4^{s-1}}+\cdots\\ \end{align}

so that $\zeta(s)\sum_{n=1}^\infty\frac{\phi(n)}{n^s}=\zeta(s-1)$. It follows that

$$\sum_{n=1}^\infty\frac{\phi(n)}{n^s}=\frac{\zeta(s-1)}{\zeta(s)}$$

and so that only converges for $s>2$.