Primes, Divisors, and Permutations

关于正整数的性质的问题，特别是关于整数的因式和素数的分布的问题，几千年来一直吸引着人们，最近在计算机科学、信息安全和信号处理中得到了应用。研究的目的是加深我们对素数的分布、整数的约数的性质、随机排列的分布的理解。一个同样重要的目标是更充分地了解这些看似不同的对象之间更深层次的联系，即素数和排列，从而在数论、组合学、概率和群论等数学领域之间建立桥梁。其中一个项目旨在开发一种新的方法，基于概率推理，建立稀疏整数序列中连续合成值的长串的存在，以便更好地了解序列素值之间的差距。特别令人感兴趣的是给定多项式的值序列。另一个项目将表明，当许多剩余类被放在一起观察时，在精确的定量意义上，在给定模的算术级数中素数的分布存在极大的差异。这还将涉及创建新的工具来处理多变量分布的大偏差。第三个项目致力于更好地理解整数的除数的集中和固定的随机排列集的集中。这项研究将改进现有关于因子在并元区间中的最大浓度分布以及任意大小固定集合的最大数目的结果，并对包含任意数量k的典型整数的k个因子的最大区间获得精确的界限。关于因子和排列的通用概率模型将指导研究。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Questions about properties of positive integers, especially the way in which integers factor and the distribution of prime numbers, have fascinated people for thousands of years and have recently found applications in computer science, information security and signal processing. Goals of the research are to enhance our understanding of the distribution of prime numbers, properties of divisors of integers, the distribution of random permutations. An equally important objective is to more fully understand the deeper connections between these seemingly dissimilar objects, that is, primes and permutations, thus building bridges between several areas of mathematics such as number theory, combinatorics, probability, and group theory.One of the projects aims to develop a new methods, based on probabilistic reasoning, for establishing the existence of long strings of consecutive composite values in sparse sequences of integers, in order to gain a better understanding of the gaps between prime values of the sequence. Of particular interest are the sequences of values of a given polynomial. Another project will show that there exist extremely large discrepancies, in a precise quantitative sense, in the distribution of primes in arithmetic progressions to a given modulus when many residue classes are viewed together. This will also involve the creation of new tools for dealing with large deviations of multivariate distributions. The third project is dedicated toward a better understanding of the the concentration of divisors of integers and the concentration of fixed sets of random permutations. The research will improve existing results about the distribution of the maximal concentration of divisors in dyadic intervals, and maximal number of fixed sets of any size, and obtain sharp bounds on the largest interval which contains k divisors of a typical integer, for any quantity k. A common probabilistic model for divisors and permutations will guide the investigations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.