If the Dirichlet series $$ \sum_{n=1}^{\infty}\frac{f(n)}{n^s} $$ converges absolutely for $\Re(s)>1$, does it follow that the Dirichlet series $$ \sum_{n=1}^{\infty}\frac{f(n^2)}{n^s} $$ also converges absolutely for $\Re(s)>1$ ?
It obviously holds when $f$ is bounded, and it also holds when $f(n)=\log(n)$. I'm unable to find a function $f$ that would not satisfy this property.
Can someone provide a counter-example?