Let $\operatorname{rad}(n)$ the product of distinct primes dividing $n>1$, with the definition $\operatorname{rad}(1)=1$ (if you need it, see the definition of the radical of an integer from this Wikipedia). Also we consider the Liouville function $\lambda(n)$ (see its definition from this MathWorld).
Question (I changed the question; I've decided it, see the comments) Is it possible to deduce if $$\sum_{1\leq n\leq x}\frac{\lambda(n)}{\operatorname{rad}(n)}\log\left(\frac{x}{n}\right)\tag{1}$$ does converge as $x\to\infty$? Thanks in advance.
I don't know if this question was in the literature, if it is known refers the literature and I try to find and read it. I've created this question as an analogous question from a Math Overflow post.
If my reasoning is right I can to find the Euler product of (I know how to calculate the Euler product and its derivative) $$\sum_{n=1}^{\infty}\frac{\lambda(n)}{\operatorname{rad}(n)}\frac{1}{n^s}\tag{2}$$ for $\Re s>\sigma$ and to find the derivative $$\sum_{n=1}^{\infty}\frac{\lambda(n)}{\operatorname{rad}(n)}\frac{\log n}{n^s}.\tag{3}$$