Define $ rad(n):=\prod_{pn}p $ and $a_n:=\frac{n}{rad(n)}.$ For example $a_n=1$ whenever n is a squarefree integer. The assosiated Dirichlet series $$F(s):=\sum_{n} \frac{a_n}{n^s}=\prod_{p} (1+\frac{1}{p^s} \frac{1}{1p^{1s}})$$ has abcissa of convergence $s_0=1.$ Are there any results regarding the distribution of $a_n$, e.g. whether $\sum \limits_{n \leq x} a_n \ll x (\log x)^A $ for some real constant $A>0$ ?
This problem was studied by De Bruijn, see
The main result is that $$\log\left(S(x)\right)=\log\left(\sum_{n\le x} \frac{1}{rad(n)}\right)\sim \left(\frac{8\log x}{\log \log x}\right)^{1/2}$$ and that $$\sum_{n\le x} \frac{n}{rad(n)}=o\left(xS(x)\right).$$ See also the article "Idempotents and Nilpotents Modulo n" for a discussion of this and similar problems (and more complete references). 

