The distribution of prime numbers and products of few primes

该项目致力于研究素数链及其应用，两个素数乘积的整数分布，整数和移位素数的除数，广义Wieferich素数以及覆盖同余的效率。 素数链是素数序列，使得p|（p '-1）对于序列中的每对连续素数p，p'，例如2，5，31。 研究了具有给定始素数、给定终素数和给定长度的素数链的分布。 应用程序给出的高度的普拉特树，并分配arithmeticfunctions。 其中一个亮点的工作是发展的一个模型的素数链与一个给定的结束素数这是基于一个概率随机碎裂过程。 我们知道在具有相同模的算术级数中素数的分布存在着微妙的不等式，我们发展了两个素数乘积的数的平行理论，强调了与素数情形的相似性和不同性，这类问题与Dirichlet L-函数的零点分布密切相关，零点的重数在我们的研究中是重要的. 我们将研究因式分解问题，如素数p的分布，使得p-1在给定的区间内有一个除数，以及整数被写为k个因子的乘积时因子的分布，对于一般的k。 我们将改进已知的最小正整数B的估计，使得p是基B的广义Wieferich素数。 一组其并集为所有整数的同余类是一个同余覆盖系。 模的倒数的和S必须至少为1，并且我们研究如果模是不同的并且大于N，S可以如何接近1。关于正整数的性质的问题，特别是整数因子和素数分布的方式，数千年来一直吸引着人们，最近在计算机科学和信息安全方面得到了应用。的项目有关的配置素数，如何数的产品是几个素数分布在算术级数，分布的除数整数，以及如何有效的覆盖系统的congruencescan。 研究项目将涉及研究生助理和博士后。

This project is dedicated to the study of prime chains and applications,the distribution of integers in progressions which are the product of two primes, divisors of integers and of shifted primes,generalized Wieferich primes, and the efficiency of covering congruences. Prime chains are sequences of primes so that p|(p'-1) for each pair of consecutive primes p, p' in the sequence, e.g.2,5,31. We study the distribution of prime chains with a given startingprime, given ending prime, and chains with a given length. Applications are given to the height of Pratt trees, and to the distribution of arithmeticfunctions. One of the highlights of the work is the development of a model of prime chains with a given ending prime which isbased on a probabilistic random fragmentation process. It is knownthat there are subtle inequities in the distribution of primes in arithmetic progressions with the same modulus, and we develop a paralleltheory for number which are products of two primes, emphasizing the similarities and differences compared with the primes case.Such problems are intimately connected with the distribution of zerosof Dirichlet L-functions, the multiplicity of zeros being important inour studies. We will investigate factorization problems, such as the distribution of primes p so that p-1 has a divisor in a given interval,and the distribution of factors when integers are written as theproduct of k factors, for general k. We will improve knownestimates for the smallest positive integer b such thatp is a generalized Wieferich prime to base b. A set of congruenceclasses whose union is all the integers is a covering system of congruences. The sum S of the reciprocals of the moduli must be at least 1,and we study how close S can be to 1 if the moduli are distinct andgreater than N.Questions about properties of positive integers, especially the way inwhich integers factor and the distribution of prime numbers, have fascinated people for thousands of years and have recently foundapplications in computer science and information security.This proposal is concerned with a number of projects aboutconfigurations of prime numbers, how number that are products of few primes are distributed in arithmetic progressions, the distribution ofdivisors of integers, and how efficient covering systems of congruencescan be. The research projects will involve graduate assistants andpost-docs.