If we know Dirichlet generating function F(s) of $f(n)$ and G(s) of g(n) we can express generating function of Dirichlet convolution of $f(n)$ and $g(n)$ as product of the two generating functions $F(s) G(s)$:
$$F(s)=\sum _{n\geq 1}{\frac {f(n)}{n^{s}}}$$ $$G(s)=\sum _{n\geq 1}{\frac {g(n)}{n^{s}}}$$ $$\sum _{n\geq 1}{\frac {(f*g)(n)}{n^{s}}}=F(s) G(s)$$
Is there anything we can say about generating function of term-wise product of $f(n)$ and $g(n)$. Is it possible to express it in terms of $F(s)$ and $G(s)$?
$$\sum _{n\geq 1}{\frac {f(n) g(n)}{n^{s}}}=???$$
No, unless you allow series or integrals. The simplest expression is certainly $$\lim_{T\to \infty}\frac1{2 T} \int_{-T}^T F(z+it)G(z-it)dt = \sum_n f(n)g(n)n^{-2z}$$ if both $F(s),G(s)$ converge uniformly on $\Re(s)=\Re(z)$.