$$x=\sum_{n=1}^\infty \frac{\phi(n)}{n^a},\quad\text{where $\phi$ is the Euler-phi/totient function and $a\geq1$}$$
Can this even be evaluated? It clearly converges for all $a>2$, since the maximum value of the totient function for any $n$ is $n-1$ and $x=\sum_{n=1}^\infty \frac{n-1}{n^a}$ converges if $a>2$.
I honestly have no idea where to begin on this one.
From Kemono Chen: This series is discussed in Chapter 3 of Tom M. Apostol's Introduction to Analytic Number Theory. Excercises 6 and 8 provide the result
$$\sum_{n=1}^\infty \frac{\phi(n)}{n^a}=\frac{\zeta(a-1)}{\zeta(a)}$$
With the implication that the series diverges for $a=2$
From Slade: While the sum is finite for $a>2$, the limit as $a\to2^+$ is $+\infty$.