Background: Once an analytic number theorist remarked to me that all attempts to prove the Riemann hypothesis using number theoretic methods have failed. Since then that remark stuck in my mind.
The veracity of the above alluded number theorist's opinion does not really matter for the question to make sense; I just included it for background.
What are some promising methods from outside number theory to approach Riemann hypothesis?
I know two:
The geometric approach of Artin, Hasse, Weil and Deligne, the most important result being the proof of the Weil Conjectures.
The Bost-Connes approach. This is outlined by Lieven Le Bruyn for instance and has a hint of thermodynamics .
I imagine that both of the above are cited by some people as the basis for the hopes that the theory of the field with one element will prove the Riemann hypothesis. Again, this question formally has no need to be connected the theory of field with one element to make sense. Other than just mentioning the above, let us not get into that aspect.
I am interested in other possible and promising methods. I am not interested in an equivalent formulation of Riemann hypothesis which is no better than the original. Both the above are very promising in terms of undiscovered things and might give a much better "big picture".
An approach I am ambivalent about, is that of Baez-Duarte. Though it does provide some evidence. I do not know whether it is any easier to prove Riemann hypothesis that way, rather than the original statement.
Please give me examples of any other methods; preferably very "promising" ones.
Edit 1: The meaning of "methods outside number theory" is the following: Nothing in the book of Ivic or Titchmarsch and Heath-Brown. More precisely, methods outside the traditional sybjects of elementary number theory and analytic number theory. I have given two examples above. One with algebraic geometry and one with thermodynamics.