Let $c_{q}(n)=\displaystyle \sum_{\substack{a=1,...,q\\ (a,q)=1}} e(\frac{an}{q})$ be the Ramanujan's sum and consider the following series: $\displaystyle\sum_{q>r} \frac{\mu^{2}(q)c_{q}(n)}{\varphi^{2}(q)}$. It's possible to find an upper bound explicitly in $n$ and $r$?

I want to find a form such as $\frac{n}{\varphi(n)}\frac{C}{r}$, with $C>0$.

I try to do this by myself, but in all my attemps I reduce me to find an explicit bound for a sum that is convergent but not absolutely convergent and so doesn't have an Euler's product (that in fact diverge).

Thanks in advance for any suggestion.

Let $c_{q}(n)=\displaystyle \sum_{\substack{a=1,...,q\\ (a,q)=1}} e\left(\frac{an}{q}\right)$ be the Ramanujan's sum and consider the following series:

$$\displaystyle\sum_{q>r} \frac{\mu^{2}(q)c_{q}(n)}{\varphi^{2}(q)}$$

It's possible to find an upper bound explicitly in $n$ and $r$?

I want to find a form such as $\frac{n}{\varphi(n)}\frac{C}{r}$, with $C>0$.

I try to do this by myself, but in all my attemps I reduce me to find an explicit bound for a sum that is convergent but not absolutely convergent and so doesn't have an Euler's product (that in fact diverge).

Thanks in advance for any suggestion.