The main attacks on the Riemann Hypothesis?

"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(http://arxiv.org/abs/math/9811068), 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 :

http://www.man.poznan.pl/cmst/2008/v_14_1/cmst_47-54a.pdf

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.

I am hesitating to write this answer since I know too little about the whole subject. However it is already a couple of years ago that I stumbled on a interesting paper on the arXiv https://arxiv.org/pdf/1703.03827.pdf by Vladimir Blinovsky .This paper seems to me like a simple and appealing idea to tackle the problem. It just seems strange for me that this idea has not attracted much publicity since it has been lying around on the arXiv for a quite long time. As a matter of fact this idea may be even described in plain words without resorting to mathematics. Let me do this now.

Let us fix the notation: \begin{equation} \zeta(s) := \sum\limits_{n=0}^\infty \frac{1}{n^s} \quad (A) \end{equation} where $s\in {\mathbb C}$ and $Re[s] > 1$. By analytic continuation (as originally carried out by Riemann https://en.wikipedia.org/wiki/File:Ueber_die_Anzahl_der_Primzahlen_unter_einer_gegebenen_Gr%C3%B6sse.pdf by deforming the integration contour in the complex plane and using Cauchy theorem) we obtain the functional equation: \begin{equation} \zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s) \quad (B) \end{equation} for $s\in {\mathbb C}$ and $s\neq1$.

Let us define the squared module of the zeta function as follows: \begin{equation} K(\sigma,T) := \left| \zeta(s)\right|^2 \end{equation} where $s=\sigma + \imath T$. Clearly since $\zeta(s)$ is continuous and smooth the same holds for the function $K$, i.e. it too is continuous and smooth.

Now comes the important part. What Blinovsky claims in his paper is that the function $K(\sigma,T)$ above is convex in the variable $\sigma$. In other words he claims that \begin{equation} \frac{\partial^2 K(\sigma,T)}{\partial \sigma^2} \ge 0 \quad (C) \end{equation} for $\sigma \in (0,1)$ and $T$ being big enough, i.e. being bigger than some number which is independent of $\sigma$. Note that the property (C) along with functional equation (B) immediately implies the Riemann hypothesis being true. Indeed let us assume that there is some stray zero off the critical line at some $(\sigma = \sigma_0,T)$ where $\sigma_0 \in (1/2,1)$. Then from the functional equation there must be another zero at $(1-\sigma,T)$ where $1-\sigma_0 \in (0,1/2)$. But this would actually mean that the set $\left\{\xi \in (1-\sigma_0,\sigma_0) | K(\xi,T)\right\}$ is either strictly negative which it cannot be since per definition the function $K$ is non-negative or that the set in question is identically equal to zero which again is impossible because the function is smooth per definition.

Now the question appears how does Blinovsky prove the convexity? He starts from a certain integral representation of the zeta function and then by changing variables appropriately and then differentiating with respect to $\sigma$ twice he ends up with a following neat expression . We have: \begin{equation} \frac{\partial^2 K(\sigma,T)}{\partial \sigma^2} = \frac{8 }{\pi T} \left(\int\limits_0^1 f(h,T) \cos\left( \frac{\pi h}{2}\right)^2 dh - \frac{1}{2} \int\limits_0^1 f(h,T) dh \right) \end{equation} where $f(h,T)$ is some complicated function expressed through an improper integral and an infinite sum, a function to long to be written down in here without the benefit of any insight. Now since $\int\limits_0^1 \cos(\pi h/2)^2 dh = 1/2$ proving the convexity property is equivalent to the Fortuin–Kasteleyn–Ginibre (FKG) inequality https://en.wikipedia.org/wiki/FKG_inequality . Since the squared cosine is monotonically decreasing in $h\in(0,1)$ the only thing one needs to do is to prove that $f(h,T)$ too is monotonically decreasing in the domain in question. Blinovsky proceeds to show that by looking at the derivative $\partial_h f(h,T)$ and proving that it is negative for all $h \in(0,1)$ and for $T$ being big enough.

Having said all this my question would be is this approach to this problem sound or instead is there anything internally flawed in here that won't make this thread worth pursuing at all.

I would really appreciate some feedback. Thank you in advance for it.