My opinion is ;

We may use id(d)=d arithmetic function and log*id dirichlet convolution in the question. i thought that ; when we multiply and divide n with $(\log d) / d$ we obtain $F(S)=\sum_{n=1}^{\infty} \frac{1}{n^{s+1}} \cdot\left(\log*id\right)$ so

$F(S)=D(log,s+1).D(id,s+1)$

So we get

$F(S)=-\zeta^{\prime}(s+1). \zeta(s)$

i am curious that my solution is right or not.You may write your solutions and different ideas. thanks for your helps.