I confess to more ignorance on the Riemann Hypothesis than on the previous topics we were discussing. I am flagging The_Reader_David for this also; I suspect that his opionions would be better informed than mine. However...even if AC is not involved, it would be possible for the Riemann Hypothesis to be formally undecidable in PA or even ZF. I don't know of anything that would prevent the RH from being undecidable, but my knowledge of undecidability is about twenty years old. I don't believe that any of the "forcing" methods that I had seen could prove its undecidability—if indeed it is undecidable—since it is (I think) first order. OTOH, when I left that field, they were looking for a method that would be to number theory as forcing was to set theory. If such a method has been found, I'm sure someone is trying to use it on RH.
Have you seen the 'proof' that the RH is true with probability one?
In other words, the relative density of possible counterexamples must be zero? No, is there a link? Thanks for the link you provided for that number theory/physics page.
Here's an idea for getting the RH proved. Find a brilliant young mathematician who owns a book that mentions the RH. Arrange an "accident" so that he dies in his early twenties. Forge a note that appears to be his handwriting in the margin of the book that says, "I have found a truly marvelous proof of this hypothesis..." etc. and arrange for the book to be found in his belongings.
Just kidding.
Here's an idea for getting the RH proved
Whenever G H Hardy took a journey he would always send Littlewood a postcard that was too small to contain the proof. Since God wouldn't let him get posthumous credit for the proof, He would insure that Hardy would get back OK. (or somehting like that)