Families and random matrix theory.
Granville, (from MSRI , spring 2002)  introductory essay on the Riemann Hypothesis and Riemann'szeta function (I.
Transactions of the American Mathematical Society, 27(4960), 1925.
As one of France's premier mathematicians, Legendre did othersignificant work, promoting the careers of Lagrange and Laplace,developing trig tables, geodesic projects, etc.
Her most famous work was the solution to the ,which has been called a "genuine highlight of 19thcentury mathematics."Other than the simplest cases solved by Euler and Lagrange,exact ("integrable") solutions to theequations of motion were unknown, so Kovalevskaya received fameand a rich prize when she solved the .
The result for randommatrices was shown to be universal, i.e.
The researchers spelled out several arguments for why the eigenvalues of their matrix are probably real, and why, in that case, the Riemann hypothesis is probably correct, but they came short of proving it. “Whether it will be difficult or easy to fill in the missing steps, at this point we cannot speculate,” said Brody. “Further work is needed to get a better feeling as to the scale of difficulty involved.”
Experts say that the new proposal is interesting, but that it’s far from certain whether the authors’ arguments about their unconventional quantum system can be made rigorous. “I would need more time to give a relevant opinion about the significance of their findings as a strategy towards the Riemann hypothesis,” said Paul Bourgade, a mathematician at New York University. In particular, Bourgade said, he would like to explore in more detail how the proposed quantum system compares with one previously proposed by Berry and Keating that has not yielded a concrete proof.
Introduction to random matrices.
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Our proposed mathematical model is building on the (Leibniz) mathematical transcendental term "". Interesting to be mentioned that also Schrödinger in (ScE1) uses the term "differential" (in german version) to explain "perception process between "subconscious" and "awareness" of human mind. The Leibniz transcendental concept of "differential = monad" means that there is no additional transcendence level added (which would be anyway a contradiction by itself), but the mathematical model becomes now applicable to all considered problem areas. The physicalmathematical modelling requirements (measurement/ observation/ test results validation) is still building on the test space L(2):
Wigner, "Random matrices in physics", 9(1967) 1123. A.V.

They do not fit into the general scheme of random matrix theory.
This talk will chart the rise of random matrix theory as a reputable means of studying Riemann's famous zeta function.

Random matrices and the Riemann zeta function
The exciting branch of modern mathematics, random matrix theory, provides the connection between the two fields.

Random Matrices and the Riemann Hypothesis  Yet …
Keating, "Random matrix theory and the Riemann zerosI: three and fourpoint correlations", , 8 (1995)1115–1131.
to that of the eigenvalues of a random matrix
The distribution of energy levels differences associated to these resonances shows the same characteristic features of random matrix theory."
S.
Riemann's zeta zeroes and eigenvalues of random matrix?
Joffily, "The Riemann Zeta Function and Vacuum Spectrum", Proceedings of Science, PoS (WC2004) 026, ] where it was suggested a "potential scattering" Hilbert–Pólya conjecture, such that the nontrivial zeros of Riemann's zeta function could be put in onetoone correspondence with the zeros of the swave Jost function for finite range potentials in the complex momenta plane, we extend our investigation to a relativistic S matrix for a Dirac particle scattering.
Riemann Zeros and Random Matrix Theory  SpringerLink
These methods and results have redirected several diverse branchesof modern mathematics including number theory, algebraic topology,and representation theory.
Riemann Zeros and Random Matrix ..
Among these are Fermat's conjecture (Lagrange's theorem)that every integer is the sum of four squares, and thefollowing:"Given any positive rationals a, b with a>b,there exist positive rationals c, d such thata^{3}b^{3} = c^{3}+d^{3}."(This latter "lemma" was investigated by Vieta and Fermat andfinally solved, with some difficulty, in the 19th century.
when Hilbert (transform) meets Riemann (hypothesis) ..
He is best known for work in combinatorics (especially Ramsey Theory)and partition calculus, but made contributionsacross a very broad range of mathematics, includinggraph theory, analytic number theory, probabilistic methods,and approximation theory.
Quantum chaos, random matrix theory, and the Riemann ζ
In addition to his famous writings on practical mathematicsand his ingenious theorems of geometry,Brahmagupta solved the general quadratic equation,and worked on number theory problems.