I am interested in the rate of decay of the sum 


$$\sum\_{n=N}^{\infty}\frac{\phi(n)}{n^{s}}$$
where $\phi$ is Euler's totient function and $s>2$ (in which case the sum converges trivially).  

Of course there's the trivial bound $\sum\_{n=N}^{\infty}\frac{\phi(n)}{n^{s}}\leq\sum\_{n=N}^{\infty}\frac{1}{n^{s-1}}\leq C(s)\frac{1}{N^{s-2}}$.  

I'm wondering if it's possible to do any better.



Rate of decay of the tail of Dirichlet series for Euler's totient function
It is impossible to do any better. We have


$$\sum\_{X\leq n<2X} \varphi(n)=\frac{9}{\pi^2}X^2+O(X \log X)$$


Therefore, for any positive $s>2$ we have


\begin{align\*}
\sum\_{n\geq N} \frac{\varphi(n)}{n^s} &\geq \sum\_{k\geq 0} 2^{-(k+1)s}N^{-s}\sum\_{2^kN\leq n<2^{k+1}N} \varphi(n) \\
&\geq C\sum\_{k\geq 0} 2^{-(k+1)s}N^{-s}2^{2k}N^2\\
&\geq C\_0(s)N^{2-s}
\end{align\*}
for some positive constants $C$ and $C\_0(s)$.


