Related papers: Asymptotic harmonic behavior in the prime number d…

A random composition of $n$ appears when the points of a random closed set $\widetilde{\mathcal{R}}\subset[0,1]$ are used to separate into blocks $n$ points sampled from the uniform distribution. We study the number of parts $K_n$ of this…

Let $s\_2(x)$ denote the number of digits "$1$" in a binary expansion of any $x \in \mathbb{N}$. We study the mean distribution $\mu\_a$ of the quantity $s\_2(x+a)-s\_2(x)$ for a fixed positive integer $a$.It is shown that solutions of the…

Let W be an affine PI algebra over a field of characteristic zero graded by a finite group G. We show that there exist $\alpha_{1},\alpha_{2}\in\mathbb{R}, \beta\in\frac{1}{2}\mathbb{Z}$, and $l\in\mathbb{N}$ such that…

We give asymptotic formulas for the number of balanced words whose slope $\alpha$ and intercept $\rho$ lie in a prescribed rectangle. They are related to uniform distribution of Farey fractions and Riemann Hypothesis. In the general case,…

We apply round-off to planar rotations, obtaining a one-parameter family of invertible maps of a two-dimensional lattice. As the angle of rotation approaches pi/2, the fourth iterate of the map produces piecewise-rectilinear motion, which…

This analysis which uses new mathematical methods aims at proving the Riemann hypothesis and figuring out an approximate base for imaginary non-trivial zeros of zeta function at very large numbers, in order to determine the path that those…

For $x\ge y>1$ and $u:= \log x/\log y$, let $\Phi(x,y)$ denote the number of positive integers up to $x$ free of prime divisors less than or equal to $y$. In 1950 de Bruijn [1] studied the approximation of $\Phi(x,y)$ by the quantity…

Let $f(n)$ be an arithmetic function with $f(1)\neq0$ and let $f^{-1}(n)$ be its reciprocal with respect to the Dirichlet convolution. We study the asymptotic behaviour of $|f^{-1}(n)|$ with regard to the asymptotic behaviour of $|f(n)|$…

We investigate the behavior of the solution to an elliptic diffraction problem in the union of a smooth set $\Omega$ and a thin layer $\Sigma$ locally described by $\varepsilon h$, where $h$ is a positive function defined on the boundary…

Under the Riemann Hypothesis, we connect the distribution of $k$-free numbers with the derivative of the Riemann zeta-function at nontrivial zeros of $\zeta(s)$. Moreover, with additional assumptions, we prove the existence of a limiting…

In this paper we study the asymptotic behavior of a random uniform parking function $\pi_n$ of size $n$. We show that the first $k_n$ places $\pi_n(1),\dots,\pi_n(k_n)$ of $\pi_n$ are asymptotically i.i.d. and uniform on $\{1,2,\dots,n\}$,…

Hooley conjectured that the variance V(x;q) of the distribution of primes up to x in the arithmetic progressions modulo q is asymptotically x log q, in some unspecified range of q\leq x. On average over 1\leq q \leq Q, this conjecture is…

In this article, we develop two types of asymptotic formulas for harmonic series in terms of single non-trivial zeros of the Riemann zeta function on the critical line. The series is obtained by evaluating the complex magnitude of an…

Two linear recurrences exhibit mirror symmetry connecting the constants $e$ and $\pi$. When parametrized, their asymptotic connection constants extend to meromorphic functions satisfying additive functional equations with rational…

For $k\geq 1$ an integer and $x\geq 1$ a real number, let $\pi_k(x)$ be the number of integers smaller than $x$ having exactly $k$ distinct prime divisors. Building on recent work of Matom\"aki and Radziwi\l\l, we investigate the asymptotic…

Two topics of the number theory are discussed in this paper. First, we prove that given each natural number $x\geq10^{3}$, we have \[ |{\rm Li}(x)-\pi(x)|\leq c\sqrt{x}\log x\texttt{ and } \pi(x)={\rm Li}(x)+O(\sqrt{x}\log x) \] where $c$…

The primorial $p\#$ of a prime $p$ is the product of all primes $q\le p$. Let pr$(n)$ denote the largest prime $p$ with $p\# \mid \phi(n)$, where $\phi$ is Euler's totient function. We show that the normal order of pr$(n)$ is $\log\log…

An algorithm is proposed for the determination of the asymptotics of a sum of a perturbation series from the given values of its coefficients in the strong-coupling limit. When applied to the \Phi^4 theory, the algorithm yields the…

We give general theorems on asymptotic normality for additive functionals of random tries generated by a sequence of independent strings. These theorems are applied to show asymptotic normality of the distribution of random fringe trees in…

The number partitioning problem can be interpreted physically in terms of a thermally isolated non-interacting Bose gas trapped in a one-dimensional harmonic oscillator potential. We exploit this analogy to characterize, by means of a…