It's impossible to say whether an approach is promising. In my view, the most interesting approach is via constructing $\mathbb{F}_1$. The wikipedia page has links to the various attempts. One hopes that after a suitable theory is constructed, one of the proofs of the function field analogue of RH will be translatable to the number field case.

http://en.wikipedia.org/wiki/Field_with_one_element

p.s. I think this is number theory and I don't agree with your professor.

It's impossible to say whether an approach is promising. In my view, the most interesting approach is via constructing $\mathbb{F}_1$. The wikipedia page has links to the various attempts. One hopes that after a suitable theory is constructed, one of the proofs of the function field analogue of RH will be translatable to the number field case.

http://en.wikipedia.org/wiki/Field_with_one_element

p.s. I think this is number theory and I don't agree with your analytic number theorist.