Does there exist an equivalent arithmetic circuit for each computable function? Does there exist an equivalent arithmetic circuit for each computable function? 


I've been trying to wrap my head around the statement above, but haven't found a counter example although I believe the statement to be false.

What has made me curious is that I've read some theorems stating that a protocol (cryptograhy protocol theory) can compute any computable function, but then requires that the function should be specified as an arithmetic circuit.
Any computable boolean function with a fixed-length input can be computed by an arithmetic circuit.  Consider any boolean function $f:\{0,1\}^n \to \{0,1\}$.  Then there exists a multivariate polynomial $p(x_1,\dots,x_n)$ such that $f(x_1,\dots,x_n) = p(x_1,\dots,x_n)$ for all $x_1,\dots,x_n$, where arithmetic is done modulo two (i.e., over the field $\mathbb{F}_2=\{0,1\}$).  Now every multivariate polynomial can be computed by a an arithmetic circuit, so $f$ can be computed by an arithmetic circuit.

In some sense the restrictions to fixed-length inputs is unavoidable, as any circuit inherently has a fixed number of inputs and a fixed number of outputs.   So once you decide to focus on boolean functions, then the statement you saw in the crypto literature is justified: any boolean function that can be computed by a circuit, can be computed by an arithmetic circuit.  And, any computable boolean function can be computed by arithmetic circuits, where we understand "computed" to mean that there is a family of arithmetic circuits, one per input length (the non-uniform model; this is unavoidable if you want to compute using circuits, as any one circuit can only have a fixed input length).