Does Claudius think Hamlet is crazy?

I think Claudius' main reason for sending Hamlet away is that he is afraid Hamlet knows what he did and is not actually crazy, just mad with vengeance. That is why he is so curious as to why Hamlet is crazy and why he sends so many people to spy on Hamlet or try to pry it out of him. Very early in the novel, I think in act 1, Claudius tells Hamlet that he wants him to stay in Denmark. That seemed weird to me at first, but looking back I think Claudius just wanted Hamlet to stay around so he could keep an eye on Hamlet. If Hamlet went away and found out what happened, he could sneak back and kill him. With Hamlet nearby he can have someone always keep watch on him. I think Claudius does not think Hamlet is crazy and that scares him.