Related papers: On the density function for the value-distribution…

We prove a central limit theorem for Birkhoff sums of the Rosen continued fraction algorithm. A Lasota-Yorke bound is obtained for general one-dimensional continued fractions with the bounded variation space, which implies quasi-compactness…

In two previous papers the second author proved some Carlson type density theorems for zeroes in the critical strip for Beurling zeta functions satisfying Axiom A of Knopfmacher. In the first of these invoking two additonal conditions were…

In this paper we complete our understanding of the role played by the limiting (or residue) function in the context of mod-Gaussian convergence. The question about the probabilistic interpretation of such functions was initially raised by…

We establish effective convergence rates in the Doeblin-Lenstra law, describing the limiting distribution of approximation coefficients arising from continued fraction convergents of a typical real number. More generally, we prove…

For any set $A$ of natural numbers with positive upper Banach density and any $k\geq 1$, we show the existence of an infinite set $B\subset{\mathbb N}$ and a shift $t\geq0$ such that $A-t$ contains all sums of $m$ distinct elements from $B$…

We generalize our recent construction of the zeros of the Riemann $\zeta$-function to two infinite classes of $L$-functions, Dirichlet $L$-functions and those based on level one modular forms. More specifically, we show that there are an…

In Arguin & Tai (2018), the authors prove the convergence of the two-overlap distribution at low temperature for a randomized Riemann zeta function on the critical line. We extend their results to prove the Ghirlanda-Guerra identities. As a…

We present a proof of the functional equation of the Riemann zeta-function or more precisely the Dirichlet eta-function, which proof seems to be new but follows almost immediately from Malmst\`en's paper ``De integralibus quibusdam…

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…

The evaluation of the Riemann zeta function at negative integers is a classical result typically obtained through analytic continuation or contour integration. In this paper, we present a novel and concise derivation of these special values…

In this paper we consider a dynamic Erd\H{o}s-R\'{e}nyi random graph with independent identically distributed edge processes. Our aim is to describe the joint evolution of the entries of a subgraph count vector. The main result of this…

We establish the theorems that give necessary and sufficient conditions for an arbitrary function defined in the unit disk of complex plane in order to has boundary values along classes of equivalencies of simple curves. Our results…

For a fixed integer $k\ge 3$ and fixed $1/2 < \sigma > 1$ we consider $$ \int_1^T |\zeta(\sigma + it)|^{2k}dt = \sum_{n=1}^\infty d_k^2(n)n^{-2\sigma}T + R(k,\sigma;T), $$ where $R(k,\sigma;T) = o(T) (T\to\infty)$ is the error term in the…

We show that the density functions of nearest neighbor spacing distributions for zeros of the real or imaginary part of the Riemann xi-function on vertical lines are described by the M-function which is appeared in value distributions of…

We develop and use some key concepts of potential theory, such as balayage and duality between measures and their potentials, to study the distribution of masses of subharmonic functions while restrictions to their growth near the boundary…

We study the limiting distributions of Birkhoff sums of a large class of cost functions (observables) evaluated along orbits, under the Gauss map, of rational numbers in $(0,1]$ ordered by denominators. We show convergence to a stable law…

In 1927 P\'olya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for the Riemann zeta function $\zeta(s)$ at its point of symmetry. This hyperbolicity has been proved for degrees $d\leq 3$. We…

A relationship between the Riemann zeta function and a density on integer sets is explored. Several properties of the examined density are derived.

In this article, we investigate the orthonormal Strichartz estimates and the convergence problem of the density function associated with $\partial_{x}^{3}+\partial_{x}^{-1}$. Firstly, when $\gamma_{0}\in\mathfrak{S}^{\beta}(\dot{H}^{s})$…

We consider Bergman spaces and variations of them in one or several complex variables. For some domains we show that in these spaces the generic function is totally unbounded and hence non - extendable. We also show that the generic…