Related papers: Probability laws related to the Jacobi theta and R…
In this paper, we are interested in the free Jacobi process starting at the unit of the compressed probability space where it takes values and associated with the parameter values $\lambda=1, \theta =1/2$. Firstly, we derive a…
The Jacobi theta-functions admit a definition through the autonomous differential equations (dynamical system); not only through the famous Fourier theta-series. We study this system in the framework of Hamiltonian dynamics and find…
This article describes a sequence of rational functions which converges locally uniformly to the zeta function. The numerators (and denominators) of these rational functions can be expressed as characteristic polynomials of matrices that…
In this article, we study the distribution of large values of the Riemann zeta function on the 1-line. We obtain an improved density function concerning large values, holding in the same range as that given by Granville and Soundararajan.
New recursion relations for the Riemann zeta function are introduced. Their derivation started from the standard functional equation. The new functional equations have both real and imaginary increment versions and can be applied over the…
Consider the first exit time of one-dimensional Brownian motion $\{B_s\}_{s\geq 0}$ from a random passageway. We discuss a Brownian motion with two time-dependent random boundaries in quenched sense. Let $\{W_s\}_{s\geq 0}$ be an other…
In this short note we study uniform approximations to the normal distributions by Jacobi theta functions. We shall show that scaled theta functions approach to a normal distribution exponentially fast.
From eigensolutions of the harmonic oscillator or Kepler-Coulomb Hamiltonian we extend the functional equation for the Riemann zeta function and develop integral representations for the Riemann xi function that is the completed classical…
It is shown that the probability density satisfies a hyperbolic equation of motion with the unique characteristic that in its many-particle form it contains derivatives acting at spatially remote regions. Based on this feature we explore…
A simple and elementary derivation of values at integer points for the Riemann's zeta and related functions is reported.
Elementary MAPLE calculations are used to support the claim of hep-th/9906240 that the ratios of theta-functions, associated with the Seiberg-Witten complex curves, provide Poisson-commuting Hamiltonians which describe the dual of the…
We consider the number ${\cal N}_{\theta_A}(\theta)$ of eigenvalues $e^{i \theta_j}$ of a random unitary matrix, drawn from CUE$_{\beta}(N)$, in the interval $\theta_j \in [\theta_A,\theta]$. The deviations from its mean, ${\cal…
We study the behavior of the Riemann zeta function on the critical line when the imaginary part of the argument is sampled by the Cauchy random walk. We develop a complete second order theory for the corresponding system of random variables…
In this paper we use operation of crossbreeding on the set of six transmutations of corresponding asymptotic complete hybrid formulas from our previous paper. We obtain in result the set of fifteen exact meta-functional equations. Every of…
We have translated fractional Brownian motion (FBM) signals into a text based on two ''letters'', as if the signal fluctuations correspond to a constant stepsize random walk. We have applied the Zipf method to extract the $\zeta '$ exponent…
It is shown that the normalized fluctuations of Riemann's zeta zeros around their predicted locations follow the Gaussian law. It is also shown that fluctuations of two zeros, $\gamma _{k}$ and $\gamma _{k+x},$ with $x\sim(\log k)^{\beta}$,…
We introduce and study a noncommutative two-parameter family of noncommutative Brownian motions in the free Fock space. They are associated with Kesten laws and give a continuous interpolation between Brownian motions in free probability…
It is well known that Brownian motion enjoys several distributional invariances such as the scaling property and the time reversal. In this paper, we prove another invariance of Brownian motion that is compatible with the time reversal. The…
We give an algebraic analog of the functional equation of Riemann's theta function. More precisely, we define a `theta multiplier' line bundle over the moduli stack of principally polarized abelian schemes with theta characteristic and…
Integral representation is one of the powerful tools for studying analytic continuation of the zeta functions. It is known that Hurwitz zeta function generalizes the famous Riemann zeta function which plays an important role in analytic…