相关论文: Gaussian Happy Numbers

Gaussian quadrature rules are a classical tool for the numerical approximation of integrals with smooth integrands and positive weight functions. We derive and expicitly list asymptotic expressions for the points and weights of Gaussian…

We show that counts of squarefree integers up to $X$ in short intervals of size $H$ tend to a Gaussian distribution as long as $H\rightarrow\infty$ and $H = X^{o(1)}$. This answers a question posed by R.R. Hall in 1989. More generally we…

In this article, we study special points of a simple random walk and a Gaussian free field, such as (nearly) favorite points, late points and high points. In section $2$, we extend results of [19] and suggest open problems for $d=2$. In…

In this paper, a simple explanation for the Goldbach Conjecture is given. We have shown that the probability of violating the conjecture not only for the prime numbers, but also for any subset of natural numbers whose distribution is…

Let $b \ge 2$ be an integer and $\xi$ an irrational real number. We establishes that, if the irrationality exponent of $\xi$ is less than $2.324 \ldots$, then the $b$-ary expansion of $\xi$ cannot be `too simple', in a suitable sense. This…

The main purpose of this paper of the paper is an explicite construction of generalized Gaussian process with function $t_b(V)=b^{H(V)}$, where $H(V)=n-h(V)$, $h(V)$ is the number of singletons in a pair-partition $V \in \st{P}_2(2n)$. This…

Let $b \geq 3$ be a positive integer. A natural number is said to be a base-$b$ Zuckerman number if it is divisible by the product of its base-$b$ digits. Let $\mathcal{Z}_b(x)$ be the set of base-$b$ Zuckerman numbers that do not exceed…

Let $b \geq 2$ be an integer and $S$ be a finite non-empty set of primes not containing divisors of $b$. For any non-dense set $A \subset [0,1)$ such that $A \cap \mathbb{Q}$ is invariant under $\times b$ operation, we prove the finiteness…

In this paper we study the $b$-ary expansions of the square roots of the function defined by the recurrence $f_b(n)=b f_b(n-1)+n$ with initial value $f(0)=0$ taken at odd positive integers $n$, of which the special case $b=10$ is often…

In this paper we extend some set theoretic concepts of numerical semigroups for arbitrary sub-semigroups of natural numbers. Then we characterized gapsets which leads to a more efficient computational approach towards numerical semigroups…

Given a positive integer $n$, the factorial base representation of $n$ is given by $n=\sum_{i=1}^ka_i\cdot i!$, where $a_k\neq 0$ and $0\leq a_i\leq i$ for all $1\leq i\leq k$. For $e\geq 1$, we define…

Following Stolarsky, we say that a natural number n is flimsy in base b if some positive multiple of n has smaller digit sum in base b than n does; otherwise it is sturdy. We develop algorithmic methods for the study of sturdy and flimsy…

Let $b \ge 2$ be an integer. We prove that the $b$-adic expansion of every irrational algebraic number cannot have low complexity. Furthermore, we establish that irrational morphic numbers are transcendental, for a wide class of morphisms.…

In a preliminary study of numerical humor, we propose the Perceived Specificity Hypothesis (PSH). The PSH states that, for nonnegative integers < 100, the funniness of a number increases with its apparent precision. A survey of 68…

We study the total mass of high points in a random model for the Riemann-Zeta function. We consider the same model as in [8], [2], and build on the convergence to 'Gaussian' multiplicative chaos proved in [14]. We show that the total mass…

The harmonic numbers and generalized harmonic numbers appear frequently in many diverse areas such as combinatorial problems, many expressions involving special functions in analytic number theory and analysis of algorithms. The aim of this…

We introduce the notions of sub Gaussian random variables in sub-linear expectation spaces. To avoid the problem caused by the existence of two different expectations, i.e., the upper expectation and the lower expectation, we divide the…

For a pair of random Gaussian integers chosen uniformly and independently from the set of Gaussian integers of norm $x$ or less as $x$ goes to infinity, we find asymptotics for the average norm of their greatest common divisor, with…

Let $f$ be a Gaussian random field on $\mathbb{R}^d$ and let $X$ be the number of critical points of $f$ contained in a compact subset. A long-standing conjecture is that, under mild regularity and non-degeneracy conditions on $f$, the…

Let $Z$ be a projective geometrically integral algebraic variety. This paper is concerned with estimating the number of rational points on $Z$ which have height at most $B$. The bounds obtained are uniform in varieties of fixed degree and…