The distribution of prime numbers and products of few primes

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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.