It has been recently mentioned by a speaker (his talk is completely not relevant to random matrix theory/RMT though) that modern statistics, especially random matrices theory, will help solving some number theoretic problems.


I was quite intrigued and ask for more explanation but the speaker himself said he did not understand the philosophy well enough but point me to a few keywords to search for, one among which is "Katz-Sarnak philosophy".


After a few searches, I figured out that all sources seem to point to [KS]. The major results in [KS] seems like saying that the class of general linear (compact) groups have the same n-level correlations. While some researchers [Kowalski][Miller] do mention that they applied KS philosophy, but what they have done seems very different...So for number theorists and experts on elliptic curves:



> 
> (1) When you refer to "Katz-Sarnak philosophy", what kind of
>  thinking/technique do you actually mean?
> 
> 
> (2) Is there a formalism/explanation of this philosophy in language of
>  RMT? I asked this because it might be helpful to understand it in this
>  perspective (at least to a probabilist).
> 
> 
> 


Any inputs are highly appreciated.


**Reference**


[KS]Katz, Nicholas M., and Peter Sarnak, eds. Random matrices, Frobenius eigenvalues, and monodromy. Vol. 45. American Mathematical Soc., 1999.[Google books](https://books.google.com/books?id=wXyOPbzvowsC&dq=Random%20matrices,%20Frobenius%20eigenvalues,%20and%20monodromy&lr=&source=gbs_navlinks_s)


[Kowalski]<http://blogs.ethz.ch/kowalski/2008/07/30/finding-life-beyond-the-central-limit-theorem/>


[Miller]Miller, Steven J. "One-and two-level densities for rational families of elliptic curves: evidence for the underlying group symmetries." Compositio Mathematica 140.4 (2004): 952-992.



What is the Katz-Sarnak philosophy?
The "Katz-Sarnak philosophy" is just the idea that statistics of various kinds for $L$-functions should, in the large scale limit, match statistics for large random matrices from some particular classical compact group. 


First you need to decide what kinds of zeros to look at: the high zeros of an individual $L$-function, the low zeros (near the real axis) in some family of related $L$-functions (e.g., low zeros in all Dirichlet $L$-functions, maybe with a restriction on the parity of the character), and so on. The choice of what sort of zeros you look at can have an effect, e.g., looking at statistics of the low zeros in a family can reveal a "symmetry" (similar statistics to a specific classical compact group) that is not evident when looking at high zeros of an individual $L$-function (universality, not distinguishing between different $L$-functions).


Next you need to normalize the (nontrivial) zeros. For example, zeros of an individual $L$-function tend to get close to each other high up the critical line, so you count the number of zeros up to height $T$ and then rescale them so they get average spacing 1. It's those rescaled zeros that you work with when computing your various statistics. 


Next you need to work with a suitable class of test functions and be able to actually carry out statistical calculations to reveal similarity with some class of random matrices.


Part of the motivation for people to look at these questions is the hope that it could suggest a spectral interpretation of the nontrivial zeros. Look up the Hilbert-Polya conjecture (Hilbert had nothing to do with it). 


You mentioned the Katz-Sarnak book in your question. Keep in mind that this book is entirely about $L$-functions for varieties over finite fields, *not* the ones like $\zeta(s)$ that are associated to number fields. The blog post and paper you mention by Kowalski and Miller are about $L$-functions over number fields, and that is perhaps why they look very different to you from the Katz-Sarnak book. Another paper where the number field case is treated head-on is Rudnick, Sarnak, "Zeros of principal $L$-functions and random matrix theory," *Duke Math. J.* 81 (1996), 269–322.
If you want to see a textbook's treatment of the relation between $L$-functions over number fields (like $\zeta(s)$ and Dirichlet $L$-functions) and random matrix theory, consider the book "An Invitation to Modern Number Theory" by Steve Miller and Ramin Takloo-Bighash. The random matrix theory is in Part 5.


