Related papers: Distribution of Normalized Zero-Sets of Random Ent…
We study the probability distribution of the number of common zeros of a system of $m$ random $n$-variate polynomials over a finite commutative ring $R$. We compute the expected number of common zeros of a system of polynomials over $R$.…
The main results of this article are asymptotic formulas for the variance of the number of zeros of a Gaussian random polynomial of degree $N$ in an open set $U \subset C$ as the degree $N \to \infty$, and more generally for the zeros of…
The purpose of this paper is to characterize the zero sets of holomorphic functions in the Nevanlinna class on a class of convex domains of infinite type in $\mathbb{C}^2$. Moreover, we also obtain $L^p$ estimates, $1 \leq p \leq \infty$,…
Numerical investigations around a transformation of Landau's formula suggest certain statistical regularities in the distribution of zeros of the Riemann zeta function.
This paper first surveys the connection of integrable systems of the Painleve type to various distribution functions appearing in Wigner-Dyson random matrix theory. A short discussion is then given of the appearance of these same…
Given several sequences of Hermitian holomorphic line bundles $\{(L_{kp}, h_{kp})\}_{p=1}^{\infty}$, we establish the distribution of common zeros of random holomorphic sections of $L_{kp}$ with respect to singular measures. We also study…
We formulate a parametrized uniformly absolutely globally convergent series of $\zeta$(s) denoted by Z(s, x). When expressed in closed form, it is given by Z(s, x) = (s -- 1)$\zeta$(s) + 1 x Li s z z -- 1 dz, where Li s (x) is the…
We introduce the notion of functionally compact sets into the theory of nonlinear generalized functions in the sense of Colombeau. The motivation behind our construction is to transfer, as far as possible, properties enjoyed by standard…
Probability distributions supported on the simplex enjoy a wide range of applications across statistics and machine learning. Recently, a novel family of such distributions has been discovered: the continuous categorical. This family enjoys…
The non-trivial zeros of the Riemann zeta function and the prime numbers can be plotted by a modified von Mangoldt function. The series of non-trivial zeta zeros and prime numbers can be given explicitly by superposition of harmonic waves.…
We review some probabilistic properties of the sum-of-digits function of random integers. New asymptotic approximations to the total variation distance and its refinements are also derived. Four different approaches are used: a classical…
W. Luo has investigated the distribution of zeros of the derivative of the Selberg zeta function associated to compact hyperbolic Riemann surfaces. In essence, the main results in Luo's article involve the following three points: Finiteness…
We study the zeros of exceptional Hermite polynomials associated with an even partition $\lambda$. We prove several conjectures regarding the asymptotic behavior of both the regular (real) and the exceptional (complex) zeros. The real zeros…
The probabilistic study of the value-distributions of zeta-functions is one of the modern topics in analytic number theory. In this paper, we study a certain probability measure related to the value-distribution of the Lerch zeta-function.…
The well known Erdos-Turan law states that the logarithm of an order of a random permutation is asymptotically normally distributed. The aim of this work is to estimate convergence rate in this theorem and also to prove analogous result for…
We obtain exact formulas for the cumulative distribution function of the variance-gamma distribution, as infinite series involving the modified Bessel function of the second kind and the modified Lommel function of the first kind. From…
A new class of distributional transformations is introduced, characterized by equations relating function weighted expectations of test functions on a given distribution to expectations of the transformed distribution on the test function's…
In this paper, we prove some normality criteria concerning transitivity of normality from one family of meromorphic functions to another which improve and generalize some recent results. We also prove some value distribution results for…
We study zeroes of Gaussian analytic functions in a strip in the complex plane, with translation-invariant distribution. We prove that the a limiting horizontal mean counting-measure of the zeroes exists almost surely, and that it is…
We consider a certain class of multiplicative functions $f: \mathbb N \rightarrow \mathbb C$ and study the distribution of zeros of Dirichlet polynomials $F_N(s)= \sum_{n\le N} f(n)n^{-s}$ corresponding to these functions. We prove that the…