Given the 2 functions $$ g(x)= \sum_{n=1}^{\infty}f\left(\frac{x}n\right)\log(n)\;, $$ how can I use Möbius inversion to recover $f$ from $g$?? I believe that
$$ f(x)= \sum_{n=1}^{\infty}\mu (n)g\left(\frac{x}n\right)\log(n)\;. $$ Here 'mu' is the Möbius function.