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"there isn't a truly compelling reason to believe that RH is true, aside from numerical evidence and a belief in beauty."

You wouldn't consider the analogue of the Riemann hypothesis for the zeta function of a smooth projective variety over a finite field, proven by Deligne, to be a compelling reason to believe the RH is true?

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No because that's a polynomial. The proofs that involve computing moments already have an analog for the Riemann zeta function, but for zeta(s) they just represent zero-density results, while in the Deligne case they give you the full on Riemann Hypothesis.

I am curious which particular proofs you are referring to, could you please give a link or reference or something?

I'm having some difficulty finding it, but Terry Tao had a nice exposition in one of his posts, I believe the crux was computing/bounding traces of powers and then being able to conclude the result. My work is almost purely on the analytic side (hence perhaps the RH skepticism), so I don't know the right reference off-hand.

Was it https://terrytao.wordpress.com/2013/07/19/the-riemann-hypoth...?

I don't get anything when I ctrl-F for "moment" or "density" there (other than in sidebar links).

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