Complexity class for Optimization problems over #P functions
It's in $\mathrm{FP^{\#P}}$, since you can compute it in poly time using oracle calls to compute the&nbsp;$f$s.  If you can poly-time Turing reduce a $\mathrm{\#P}$-complete problem to it, then $\tilde{f}$&nbsp;is $\mathrm{\#P}$-hard but you can't say it's $\mathrm{\#P}$-complete unless you can show that it's in&nbsp;$\mathrm{\#P}$. That seems unlikely, to me, as I can't see any way of building a nondeterministic Turing machine to have that many accepting paths.

Since $\tilde{f}\in\mathrm{FP^{\#P}}$ and $\mathrm{FP^{FP^{\#P}}}=\mathrm{FP^{\#P}}$ (just compose the reductions), $\mathrm{FP}^{\tilde{f}}\subseteq \mathrm{FP^{\#P}}$.  Note that $\mathrm{FP}^{\tilde{f}}$ is a complexity class (all problems that are poly-time Turing reducible to&nbsp;$\tilde{f}$), not a problem.