I have an easy problem with Turing Machines: Let $f$ be a function in $\#P.$ Then there is some polynomial-time TM $M$ such that for every input $x, f (x)$ is the number $\#M(x)$ of strings $u ∈ \{0,1\}^m$ such that $M(x, u) = 1$, where $m$ is some polynomial in $|x|$ that is the length of certificates that $M$ takes.
For every two TM’s $M_0,M_1$ taking $m$-bit certificates, denote in this proof by “$M_0+M_1$” the TM M' that takes $n + 1$ bit certificate where $M' (x, bu) = M_b(x, u)$. Then $\#_{M_0+M_1} (x) = \#M_0 (x) + \#M_1 (x)$. WHY ?? Also, for $N ∈ \{0 . . . 2^m\}$, we denote by $M_N$ the TM that on input $x,u$ outputs $1$ iff the string $u$, when considered as a number, is smaller than $N$. Clearly $\#M_N (x) = N$. Again, WHY??