The distribution of prime numbers and products of few primes

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

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 starting prime, given ending prime, and chains with a given length. Applications are given to the height of Pratt trees, and to the distribution of arithmetic functions. One of the highlights of the work is the development of a model of prime chains with a given ending prime which is based on a probabilistic random fragmentation process. It is known that there are subtle inequities in the distribution of primes in arithmetic progressions with the same modulus, and we develop a parallel theory 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 zeros of Dirichlet L-functions, the multiplicity of zeros being important in our 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 the product of k factors, for general k. We will improve known estimates for the smallest positive integer b such that p is a generalized Wieferich prime to base b. A set of congruence classes 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 and greater than N. 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 and information security. This proposal is concerned with a number of projects about configurations of prime numbers, how number that are products of few primes are distributed in arithmetic progressions, the distribution of divisors of integers, and how efficient covering systems of congruences can be. The research projects will involve graduate assistants and post-docs.