Related papers: Some experiments in number theory
We announce a number of conjectures associated with and arising from a study of primes and irrationals in $\mathbb{R}$. All are supported by numerical verification to the extent possible.
Large random matrices appear in different fields of mathematics and physics such as combinatorics, probability theory, statistics, operator theory, number theory, quantum field theory, string theory etc... In the last ten years, they…
These expository notes are centered around the circular law theorem, which states that the empirical spectral distribution of a nxn random matrix with i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the complex…
In this paper we present some observations about the well-known Goldbach conjecture. In particular we list and interpret some numerical results which allow us to formulate a relation between prime numbers and even integers. We can also…
We provide a perturbative expansion for the empirical spectral distribution of a Hermitian matrix with large size perturbed by a random matrix with small operator norm whose entries in the eigenvector basis of the first one are independent…
Prime numbers have attracted the attention of mathematiciansand enthusiasts for millenniums due to their simple definition and remarkable properties. In this paper, we study primorial numbers (the product of the first prime numbers) to…
We prove the Central Limit Theorem for the number of eigenvalues near the spectrum edge for hermitian ensembles of random matrices. To derive our results, we use a general theorem, essentially due to Costin and Lebowitz, concerning the…
We prove multi-dimensional central limit theorems for the spectral moments (of arbitrary degrees) associated with random matrices with real-valued i.i.d. entries, satisfying some appropriate moment conditions. Our techniques rely on a…
We consider a random matrix whose entries are independent Gaussian variables taking values in the field of quaternions with variance $1/n$. Using logarithmic potential theory, we prove the almost sure convergence, as the dimension $n$ goes…
We study the spectral measure of large Euclidean random matrices. The entries of these matrices are determined by the relative position of $n$ random points in a compact set $\Omega_n$ of $\R^d$. Under various assumptions we establish the…
We will see that key concepts of number theory can be defined for arbitrary operations. We give a generalized distributivity for hyperoperations (usual arithmetic operations and operations going beyond exponentiation) and a generalization…
Random matrix theory is a well-developed area of probability theory that has numerous connections with other areas of mathematics and its applications. Much of the literature in this area is concerned with matrices that possess many exact…
Random matrices whose entries come from a stationary Gaussian process are studied. The limiting behavior of the eigenvalues as the size of the matrix goes to infinity is the main subject of interest in this work. It is shown that the…
Consider the ensemble of real symmetric Toeplitz matrices, each independent entry an i.i.d. random variable chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous investigations showed that…
Ova-angular rotations of a prime number are characterized, constructed using the Dirichlet theorem. The geometric properties arising from this theory are analyzed and some applications are presented, including Goldbach's conjecture, the…
Consider Ginibre's ensemble of $N \times N$ non-Hermitian random matrices in which all entries are independent complex Gaussians of mean zero and variance $\frac{1}{N}$. As $N \uparrow \infty$ the normalized counting measure of the…
These lectures provide an informal introduction into the notions and tools used to analyze statistical properties of eigenvalues of large random Hermitian matrices. After developing the general machinery of orthogonal polynomial method, we…
In order to have a better understanding of finite random matrices with non-Gaussian entries, we study the $1/N$ expansion of local eigenvalue statistics in both the bulk and at the hard edge of the spectrum of random matrices. This gives…
We explain how Gaussian integrals over ensemble of complex matrices with source matrices generate Hurwitz numbers of the most general type, namely, Hurwitz numbers with arbitrary orientable or non-orientable base surface and arbitrary…
The classical random matrix theory is mostly focused on asymptotic spectral properties of random matrices as their dimensions grow to infinity. At the same time many recent applications from convex geometry to functional analysis to…