Related papers: The aliquot constant

We study the evolution of the susceptibility in the subcritical random graph $G(n,p)$ as $n$ tends to infinity. We obtain precise asymptotics of its expectation and variance, and show it obeys a law of large numbers. We also prove that the…

We show the following result: Assume B is an infinite Boolean Algebra and lambda=d(B). Then s(B*B)$, i.e. s(uf(B)xuf(B))>= lambda$ (if lambda limit - obtained)

Let x be a quadratic irrational and let P be the set of prime numbers. We show the existence of an infinite subset S of P such that the statistics of the period of the continued fraction expansions along the sequence {px: p\in S} approach…

Let X_1,X_2, . . . be a sequence of i.i.d. mean zero random variables and let S_n the sum of the first n random variables. We show that whenever lim sup_n |S_n|/c_n is finite with probability one and the normalizing sequence {c_n} is…

Prime numbers are one of the most intriguing figures in mathematics. Despite centuries of research, many questions remain still unsolved. In recent years, computer simulations are playing a fundamental role in the study of an immense…

The paper is primarily concerned with the asymptotic behavior as $N\to\infty$ of averages of nonconventional arrays having the form $N^{-1}\sum_{n=1}^N\prod_{j=1}^\ell T^{P_j(n,N)}f_j$ where $f_j$'s are bounded measurable functions, $T$ is…

The goal of this article is to study the discrepancy of the distribution of arithmetic sequences in arithmetic progressions. We will fix a sequence $\A=\{\a(n)\}_{n\geq 1}$ of non-negative real numbers in a certain class of arithmetic…

We consider the sums $T_N=\sum_{n=1}^N F(S_n)$ where $S_n$ is a random walk on $\mathbb Z^d$ and $F:\mathbb Z^d\to \mathbb R$ is a global observable, that is, a bounded function which admits an average value when averaged over large cubes.…

For every 3/4 <= beta < 1 we construct a finitely generated group so that the expected distance of the simple random walk from its starting point is within a constant factor of n^beta. In fact, the speed can be set precisely to equal any…

This paper explores the joint behaviour of the summands of a random walk when their mean value goes to infinity as its length increases. It is proved that all the summands must share the same value, which extends previous results in the…

In this note we consider a multi-slit Loewner equation with constant coefficients that describes the growth of multiple SLE curves connecting $N$ points on $\mathbb{R}$ to infinity within the upper half-plane. For every $N\in\mathbb{N}$,…

The Mallows measure on the symmetric group $S_n$ is the probability measure such that each permutation has probability proportional to $q$ raised to the power of the number of inversions, where $q$ is a positive parameter and the number of…

We prove prime exponential sums have no better than square root cancellation on average on short intervals, in the sense that $$\frac{1}{x} \sum_{-y< n\le x} \left|\sum_{\substack{n< m \le n+y\\ 1\le m \le x}} \Lambda(m) \mathrm{e}(\alpha…

Let U(N) denote the maximal length of arithmetic progressions in a random uniform subset of {0,1}^N. By an application of the Chen-Stein method, we show that U(N)- 2 log(N)/log(2) converges in law to an extreme type (asymmetric)…

Let $R(n)$ denote the number of rich words of length $n$ over a given finite alphabet. In 2017 it was proved that $\lim_{n\rightarrow\infty} \sqrt[n]{R(n)}=1$; it means the number of rich words has a subexponential growth. However, up to…

This paper concerns the asymptotic behavior of a random variable $W_\lambda$ resulting from the summation of the functionals of a Gibbsian spatial point process over windows $Q_\lambda \uparrow R^d$. We establish conditions ensuring that…

Let $\Sigma$ be a countable alphabet. For $r\geq 1$, an infinite sequence $s$ with characters from $\Sigma$ is called $r$-quasi-regular, if for each $\sigma\in\Sigma$ the ratio of the longest to shortest interval between consecutive…

A continuous approximation for the results of [1] is obtained. In this approximation the energy distribution is represented in the form of the product of the Gibbs factor and superstatistics factor. The mutual weights of the factors are…

Let H(N) denote the set of all polynomials with positive integer coefficients which have their zeros in the open left half-plane. We are looking for polynomials in H(N) whose largest coefficients are as small as possible and also for…

We study mean convergence of multiple ergodic averages, where the iterates arise from smooth functions of polynomial growth that belong to a Hardy field. Our results include all logarithmico-exponential functions of polynomial growth, such…