Recall that the radical of an integer $n$ is defined to be $\operatorname{rad}(n) = \prod\_{p \mid n } p$. 


For a paper, I need the result that
$$\sum\_{n \leq x} \frac{1}{\operatorname{rad}(n)} \ll\_\varepsilon x^{\varepsilon} \tag{$\*$},$$
for all $\varepsilon > 0$. I have a proof of this using complex analysis and Perron's formula, but this seems a bit overkill given that I'm looking for a weak upper bound for a problem in elementary number theory.



> 
> Does anyone know of a short elementary proof of the bound $(\*)$? Or better yet, a reference?
> 
> 
> 



Sum of the reciprocals of radicals
You can get away with elementary analytic number theory. Consider the series $\sum\_n\frac{1}{n^{\varepsilon}\rm{rad}(n)}$. It suffices to show that it converges. However, it can be written as a product of 
$$
S(p)=1+p^{-1-\varepsilon}+p^{-1-2\varepsilon}+\dots=1+p^{-1-\varepsilon}\frac 1{1-p^{-\varepsilon}}\le 1+p^{-1-\frac\varepsilon 2}
$$
for all but finitely many $p$.
Thus $\prod\_p S(p)\le C\prod\_p(1+p^{-1-\frac\varepsilon 2})\le\sum\_n n^{-1-\frac\varepsilon 2}<+\infty$ 


