how to solve this equation for $ f(x) $ in a functional equation

53 Views Asked by At

given the functional infinite equation

$$ g(x)= \sum_{n=2}^{\infty}f(\frac{x}{n})log(n) $$

the function $ g(x) $ is know

if the log term wasn't there i know the solution but can i really invert this equation and obtain $ f(x) $ ?

thanks

Applying the Mellin transform to both sites i get ·$ G(s)= -\zeta ' (s)F(s) $

what can i do which math should i use ?