Related papers: The behaviour of a certain additive function in la…
Necessary and sufficient conditions under which two real functions defined on the real interval can be separated by a polynomial are given. An immediate consequence of the main result is the existence of the polynomial separation of convex…
In this article, we prove a theorem \`a la Mauduit et Rivat (prime number theorem, Moebius randomness principle) for functions that count digital blocks whose length is a growing function tending to infinity. These sequences are not…
The law of large numbers for the empirical density for the pairs of uniformly distributed integers with a given greatest common divisor is a classic result in number theory. In this paper, we study the large deviations of the empirical…
We bring to bear an empirical model of the distribution of twin primes and produce two distinct results. The first is that we can make a quantitative probabilistic prediction of the occurrence of gaps in the sequence of twins within the…
In this paper, we present a large-deviation theory developed for functionals of canonical Gibbs processes, i.e., Gibbs processes with respect to the binomial point process. We study the regime of a fixed intensity in a sequence of…
Building on the theory of causal discovery from observational data, we study interactions between multiple (sets of) random variables in a linear structural equation model with non-Gaussian error terms. We give a correspondence between…
An initial screening experiment may lead to ambiguous conclusions regarding the factors which are active in explaining the variation of an outcome variable: thus adding follow-up runs becomes necessary. We propose a fully Bayes objective…
In the present work the existence of some patterns of primes is shown which generalize the celebrated result of Green and Tao according to which there are arbitrarily long arithmetic progressions in the sequence of primes
We reveal a relationship between the prime counting function and an operation performed on a unique subsequence of the primes.
We investigate Sarnak's conjecture on the M\"obius function in the special case when the test function is the indicator of the set of integers for which a real additive function assumes a given value.
The aim of this note is to provide an upper bound of the number of positive integers $\le x$ which can be written as $\varphi(n)$ for some positive integer $n$, where $\varphi$ stands for the Euler's function. The order of magnitude of this…
In this paper we study the problem of detecting prime numbers between all consecutive cubes. Firstly, we use a large computation to show that there is always a prime between $n^3$ and $(n+1)^3$ for $n^3\leq 1.649\cdot 10^{40}$. In addition,…
A boundary value problem for a stationary nonlinear dispersive equation of order $2l+1\;\; l\in \mathbb{N}$ with a convective term in the form $u^ku_x\;\; k\in \mathbb{N}$ was considered on an interval $(0,L)$. The existence, uniqueness and…
We present the pattern underlying some of the properties of natural numbers, using the framework of complex networks. The network used is a divisibility network in which each node has a fixed identity as one of the natural numbers and the…
We investigate ergodic-theoretical quantities and large deviation properties of one-dimensional intermittent maps, that have not only an indifferent fixed point but also a singular structure such that the uniform measure is invariant under…
We numerically investigated the dynamics of a mixture of finite-size active and passive particles in a linear array of convection rolls. The interplay of advection and steric interactions produces a number of interesting effects, like the…
Distinguir entre enteros con un n\'umero par o impar de divisores primos es una de las tareas m\'as dif\'iciles en la teor\'ia anal\'itica de n\'umeros. Un trabajo reciente de Matom\"aki y Radziwi{\l}{\l} muestra que, en promedio, ambos…
The pentagonal number theorem is extended to the sequence of the number of integer partitions with all parts equal. The new pentagonal number theorem implies that the distribution of the primes is just a specific detail of the application…
We consider the summatory function of the number of prime factors for integers $\leq x$ over arithmetic progressions. Numerical experiments suggest that some arithmetic progressions consist more number of prime factors than others. Greg…
The paper studies the asymptotic behaviour of weighted functionals of long-range dependent data over increasing observation windows. Various important statistics, including sample means, high order moments, occupation measures can be given…