[FOM] Query Concerning Excluded Middle & CH

[John Burgess]

There has been at least one theorem in number theory proved by  
showing that it follows from the Riemann hypothesis AND that it  
follows from the negation of the Riemann hypothesis, and then  
appealing to excluded middle. (See Ireland & Rosen, _A Classical  
Introduction to Modern Number Theory_ , Theorem 20.6.1.) My highly  
fallible memory tells me  that 35 years or so ago, as a post-doc in  
Madison, I was told by some authority that there was a similar  
example in analysis involving CH. Can anyone confirm this --- I mean,  
confirm that there is such an example, and not confirm that I was  
told there was --- and provide a reference? (Further examples from  
number theory involving RH would also be welcome.)