I already mentioned the field with one element in thecquestion itself and it was with the hopes that the answers will not spend time on that topic.

I just asked for an approach that a person answering might think to be promising. We do not want to hear just every failes attempt, do we? I do not want to give the open problem tag because I am uncomfortable with the connotation that I am asking to prove the Riemann hypothesis.

@Tom Smith: Yes. Though I didn't mention it explicitly, what the professor told me was that they failed completely hopelessly. Whereas the non-number theoretic approaches require some theory-building and there is hope yet and for the meantime we try to do the groundwork.

@David Hansen: I mean that even if they aren't successful they might unearth a lot of interesting math. Indeed, the proof of Weil conjectures is already a lot of interesting math.

If you mean Random matrices, no, it is a separate approach.

It's not my cardinality proof; it is lifted from Rudin's book.

Yes, your formulation is the correct one. Mine was too vague and without making precise in this way it does not make sense.

@Gerald Edgar: What I had in mind when I linked, was the example of Hamel basis which I mentioned there. Yes, it is true that without AC the proof that the Borel sigma algebra has cardinality that of the real line, would fail.

@Charles Staats : Yes, that is what I mean. I have edited to incorporate this.