I am experimenting with k-SAT. I'm using an oracle that returns the total number of satisfiable truth assignments, which is in #P. The interest here is that this total is returned modulo a natural number, say N. There is therefore a 1/N probability that the oracle mistakes a satisfiable problem for one with zero satisfiable truth assignments. I'm wondering if this oracle could be non-relativizing. I'm also wondering if there has been research concerning oracles with similar types of "errors".
When considering variations of a computation model, let alone one that is not natural and might confuse our intuition, definitions are of the essense. Your oracle might be in #P but that doesn't mean that is representative of that class. Since you have changed the definition of the function the oracle computes, you will have to either prove the new problem #P-complete or find another appropriate class for which the problem is complete.
Fooling around the complexity zoo, it seems that such a class has already been defined: http://qwiki.stanford.edu/index.php/Complexity_Zoo:M#modp and http://qwiki.stanford.edu/index.php/Complexity_Zoo:M#modkp , depending on how you choose N.