The main attacks on the Riemann Hypothesis?

Attempts to prove the Riemann Hypothesis

So I’m compiling a list of all the attacks and current approaches to Riemann Hypothesis. Can anyone provide me sources (or give their thoughts on possible proofs of it) on promising attacks on Riemann Hypothesis?

My current understanding is that the field of one element is the most popular approach to RH.

It would be good if someone started a Polymath project with the aim of proving RH. Surely, if everyone discussing possible ways to prove RH, it would be proven in about a year or so or at least people would have made a bit more progress towards the proof or disproof.

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“My current understanding is that the field of one element is the most popular approach to RH.”

Analytic number theory, with ideas from algebraic geometry, random matrix theory, and any other areas that might be relevant, is the only approach known to have produced any concrete results toward RH. The random matrix theory in particular has produced a lot of new constraints and specific, provable ideas about the distribution of zeros on the critical line.

The field of one element is, for now, a speculative area of algebraic geometry whose foundations are not set. It is more an inspiration for research on more definite mathematical objects (e.g., is there a tensor product of zeta functions) than a well-defined topic of research in itself.

(I’ll add here some response to the comments. Research on $F_1$ is, as Matt E writes, “serious” and conducted with various sophisticated intentions in mind, such as perfecting the analogies between number theory and geometry, proving the Riemann hypothesis, understanding quantum groups, or realizing parts of combinatorics as geometry over $F_1$. This was all proposed in Manin’s lectures at Columbia 20 years ago which were instrumental in bringing the idea into fashion in recent years. However, as serious and sophisticated as this research is, the idea that a suitable notion of $\Bbb{F_1}$ exists as a deeper base for algebraic geometry, or that this line of research can be developed to cover new varieties beyond the original example of Weyl groups of reductive groups (or flag varieties and other examples with simple $q$-enumerations) — or the hope that all this can help prove the Riemann hypothesis — is a speculative enterprise and one whose foundations have not been established.)

It should be worth pointing out that, Alain Connes attacked the problem from a very different plane(, following Weil and Haran’s path, he tried to construct an “index theory” in Arithmetical context linking Arithmetical data with Spectral properties of a certain operator which is closed and unbounded and whose spectrum consists of imaginary parts of the zeroes of Hecke’s L function with Grossen-character. Essentially he reconstructed a theory similar to Selberg’s, he found a trace formula equivalent to the RH using Weil’s explicit formulae.

Actually Shai Haran also stated a “similar” trace formula in his AMS paper “On Riemann’s zeta function”, One can also find a derivation of his trace formula in his book The Mysteries of the Real Prime.

Also check out the following paper, on the Baez-Duarte Criterion :

In particular, look at that graph of Fig 2 on Page 5 of 8.

If someone can prove that it stays an obvious cosine function, as the associated formulae suggest, with non-increasing amplitude and no new trends creeping for larger n to throw it out, then perhaps that would be enough to prove RH !

An attack by physics based on an inverse spectral problem yields that the inverse of the potential is $V^{-1}(x) = 2\sqrt \pi \frac{d^{1/2}}{dx^{1/2}} \operatorname{Arg} \xi (1/2+i \sqrt x)$.

In this case also the Theta functions classical and semiclassical are almost equal $$ \Theta (t)= \sum_n \exp(-t E_n) = \iint_C \,dp \, dx \, \exp(-tp^2 – tV(x)) .$$ This is one of the best approximation to RH.

Now that this thread has been resurrected a few days ago, one could add Deninger’s approach using foliated spaces. Some references are

Deninger, Christopher.
On the nature of the “explicit formulas” in analytic number theory—a simple example. Number theoretic methods (Iizuka, 2001), 97–118,
Dev. Math., 8, Kluwer Acad. Publ., Dordrecht, 2002.

Leichtnam, Eric.
On the analogy between arithmetic geometry and foliated spaces.
Rend. Mat. Appl. (7) 28 (2008), no. 2, 163–188.