Related papers: Jost function, prime numbers and Riemann zeta func…

We study complex eigenvalues of large $N\times N$ symmetric random matrices of the form ${\cal H}=\hat{H}-i\hat{\Gamma}$, where both $\hat{H}$ and $\hat{\Gamma}$ are real symmetric, $\hat{H}$ is random Gaussian and $\hat{\Gamma}$ is such…

We introduce a new generalization of Stirling numbers of the second kind and analyze their properties, including generating functions, integral representations, and recurrence relations. These numbers are used to approximate Riemann zeta…

In these notes, we describe the recent progress in understanding the zero sets of two remarkable Gaussian random functions: the Gaussian entire function with invariant distribution of zeroes with respect to isometries of the complex plane,…

We prove explicit asymptotics for the location of semiclassical scattering resonances in the setting of $h$-dependent delta-function potentials on $\mathbb{R}$. In the cases of two or three delta poles, we are able to show that resonances…

We study the distribution of the zeros of functions of the form $f(s)=h(s) \pm h(2a-s)$, where $h(s)$ is a meromorphic function, real on the real line, $a$ a real number. One of our results establishes sufficient conditions under which all…

The non-trivial zeros of the Riemann zeta function are central objects in number theory. In particular, they enable one to reproduce the prime numbers. They have also attracted the attention of physicists working in Random Matrix Theory and…

We investigate the distribution of the Riemann zeta-function on the line $\Re(s)=\sigma$. For $\tfrac 12 < \sigma \le 1$ we obtain an upper bound on the discrepancy between the distribution of $\zeta(s)$ and that of its random model,…

We investigate the relationship between the maximum of the zeta function on the 1-line and the maximal order of $S(t)$, the error term in the number of zeros up to height $t$. We show that the conjectured upper bounds on $S(t)$ along with…

We investigate the intersections of the curve $\mathbb{R}\ni t\mapsto \zeta({1\over 2}+it)$ with the real axis. We show that if the Riemann hypothesis is true, the mean-value of those real values exists and is equal to 1. Moreover, we show…

Electron polarimeters based on Mott scattering are extensively used in different fields in physics such as atomic, nuclear or particle physics. This is because spin-dependent measurements gives additional information on the physical…

This paper studies zeta functions of the form $\sum_{n=1}^{\infty} \chi(n) n^{-s}$, with $\chi$ a completely multiplicative function taking only unimodular values. We denote by $\sigma(\chi)$ the infimum of those $\alpha$ such that the…

In the framework of a random matrix description of chaotic quantum scattering the positions of $S-$matrix poles are given by complex eigenvalues $Z_i$ of an effective non-Hermitian random-matrix Hamiltonian. We put forward a conjecture on…

This paper is a summary of the general approach outlined in my previous papers toward proving the riemann hypothesis. Numerical and graphical proof of the Riemann Hypothesis is presented with analytical arguments although more work needs…

The paper reviews existing results about the statistical distribution of zeros for the three main types of zeta functions: number-theoretical, geometrical, and dynamical. It provides necessary background and some details about the proofs of…

A dynamic scheme basing on equation for T-matrix momentum transfer spectral density and integral representation for Jost function is proposed for local Dirac Hamiltonians in arbitrary N- dimension spaces and for Schrodinger one with…

Fix $\delta\in(0,1]$, $\sigma_0\in[0,1)$ and a real-valued function $\varepsilon(x)$ for which $\limsup_{x\to\infty}\varepsilon(x)\le 0$. For every set of primes ${\mathcal P}$ whose counting function $\pi_{\mathcal P}(x)$ satisfies an…

We consider a two-dimensional analogue of Helmholtz resonator with walls of finite thickness in the critical case when there exists an eigenfrequency equalling to the limit of poles generated by both the bounded component of the resonator…

We describe a construction of random meromorphic functions with prescribed simple poles with unit residues at a given stationary point process. We characterize those stationary processes with finite second moment for which, after…

We investigate the intersections of the curve $\mathbb{R}\ni t\mapsto \zeta({1\over 2}+it)$ with the real axis. We show unconditionally that the zeta-function takes arbitrarily large positive and negative values on the critical line.

We numerically study the statistical properties of differences of zeros of Riemann zeta function and L-functions predicted by the theory of the e\~ne product. In particular, this provides a simple algorithm that computes any non-real…