TITLE: Proof (claimed) for Riemann hypothesis on ArXiv
QUESTION [49 upvotes]: Has anyone noticed the paper On the zeros of the zeta function and eigenvalue problems by M. R. Pistorius, available on ArXiv?
The author claims a proof of RH, and also a growth condition on the zeros.
It was posted two weeks ago, and I expected it would have been shot down by now. Has there been any discussion or attempt at verification of this preprint?

REPLY [51 votes]: I had a go reading through the paper and I think I found the error. The main argument in the paper can be summarized as follows:

The Riemann $\Xi$-function $\Xi(t) = \xi\left(\frac{1}{2} + it\right)$ satisfy $\Xi(t) = \Xi(0)\nu_t(\pi/2)$ where $$\nu_t(x) = \int_0^\infty \cos(t(y+\cos(x))\Phi(y){\rm d}y$$ and $\Phi$ is related to the Jacobi $\theta$-function. This is a result by Riemann and holds true. The author then notes that when $t$ is such that $\Xi(t) = 0$ then $\nu_t(x)$ satisfy the Sturm–Liouville (SL) problem 
  $$\left(\frac{\nu_t'(x)}{\sin(x)}\right)' + t^2\sin(x)\nu_t(x) = 0,~~~\nu_t'(0) = 0,~~~\nu_t(\pi/2) = 0$$
  This is also true. The proof is completed by appealing to a theorem that says that this problem only has real eigenvalues. If this holds then it follows that $\Xi(t) = 0\implies t\in\mathbb{R}$ which is the Riemann hypotesis.

The error is in the last step. It is indeed true that a regular SL problem only has real eigenvalues, however this is not a regular SL problem as $\frac{1}{\sin(x)}$ has a pole at $x=0$. In this case there is no guarantee that the eigenvalues have to be real and we can show this explicitly with a simple counter-example: the function $\nu_t(x) = \sin(t\cos(x))$ satisfy the SL problem above for any complex number $t$.

Given how much interest this question has generated, which I take to mean that many people thought (or perhaps just hoped) this was a potentially viable proof, I think it’s useful to talk a bit about why it had to be wrong. Personally I have only been in academia for $\sim 10$ years, but I have already managed to see $\sim 50$ papers$^1$ like this where a major result is proven in a few pages using elementary methods. This is just another one. One soon learns that papers like this are never correct and the reason is often this: if it could have been solved this way then it would have been solved this way many many years ago - the techniques used are just too simple. As for some useful pointers for how to judge for yourself if a paper like this has the potential for being correct I reccommend Scott Aaronsons "Ten Signs a Claimed Mathematical Breakthrough is Wrong".
$^1$ For some examples see the MSE questions [1], [2], [3]