相关论文: The distribution of the primes

This is an expanded account of three lectures on the distribution of prime numbers given at the Montreal NATO school on equidistribution.

Let $\{\cdot\}$ denote the fractional part and $n \geq 1$ be a fixed integer. In this short note, we show for any prime $p$ the one-to-one correspondence $$\sum_{\nu \geq 1} \left\{\frac{n}{p^\nu}\right\} > 1 \quad \iff \quad p \mid…

It is known that there are infinitely-many prime numbers which take the form of a polynomial of degree one with integer coefficients, this is Dirichlet's theorem. We use an elementary sieving argument together with bounds on the prime…

We consider the problem of sequencing a set of positive numbers. We try to find the optimal sequence to maximize the variance of its partial sums. The optimal sequence is shown to have a beautiful structure. It is interesting to note that…

In this work we state a Theorem on number theory and apply it to solve some ordinary and partial differential equations.

A hypothesis is put forward regarding the function $\pi_2(x)$ which describes the distribution of twin primes in the set of natural numbers. The function $\pi_2(x)$ is tested by evaluation and an empirical $\pi_2^{\ast}(x)$ is arrived at,…

Polynomials are common algebraic structures, which are often used to approximate functions including probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of…

Denote by {$\times$} the fractional part. We establish several new metrical results on the distribution properties of the sequence ({x n }) n$\ge$1. Many of them are presented in a more general framework, in which the sequence of functions…

This paper completes the classification of maximal unrefinable partitions, extending a previous work of Aragona et al. devoted only to the case of triangular numbers. We show that the number of maximal unrefinable partitions of an integer…

I give some claims on primorial prime numbers for interested readers in number theory.

In this paper, we use a simple discrete dynamical model to study integer partitions and their lattice. The set of reachable configurations of the model, with the order induced by the transition rule defined on it, is the lattice of all…

In this paper we describe a theory of a cumulative distribution function on a space with an order from a probability measure defined in this space. This distribution function plays a similar role to that played in the classical case.…

For each integer $x$, the $x$-th generalized pentagonal number is denoted by $P_5(x)=(3x^2-x)/2$. Given odd positive integers $a,b,c$ and non-negative integers $r,s$, we employ the theory of ternary quadratic forms to determine when the sum…

In this short note we present a class of conjectures on partitions of integers as summations of primes, which are extensions of Goldbach conjecture.

The partition algebras are algebras of diagrams (which contain the group algebra of the symmetric group and the Brauer algebra) such that the multiplication is given by a combinatorial rule and such that the structure constants of the…

A simple method to derive parametric analytical extensions of Benford's law for first digits of numerical data is proposed. Two generalized Benford distributions are considered, namely the two-sided power Benford distribution and the new…

Translation from the Latin of Euler's "Observatio de summis divisorum" (1752). E243 in the Enestroem index. The pentagonal number theorem is that $\prod_{n=1}^\infty (1-x^n)=\sum_{n=-\infty}^\infty (-1)^n x^{n(3n-1)/2}$. This paper assumes…

The number of partitions of n into parts divisible by a or b equals the number of partitions of n in which each part and each difference of two parts is expressible as a non-negative integer combination of a or b. This generalizes…

We give a new elementary proof of Landau's Prime Ideal Theorem. The proof is an extension of Richter's proof of the Prime Number Theorem. The main result contains other results related to the equidistribution of the prime ideal counting…

Physical and mathematical applications of fractional Poisson probability distribution have been presented. As a physical application, a new family of quantum coherent states has been introduced and studied. As mathematical applications, we…