Primes, Divisors, and Permutations

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

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.