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A standard use of the Z-transform ($F(z) = \sum_n (f[n] \cdot z^{-n} )$) is to understand the effect of a difference equation on a signal. For instance: $y[n] = x[n] + y[n-1]$ $Y(z) = X(z) + Y(z) \...

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=...