This proposal is concerned with three projects related to the distribution of prime numbers and divisors of integers. The first project concerns continuing investigations (jointly with S. Konyagin) of subtle inequities in the distribution of primes in arithmetic progressions. Such problems are intimately connected with the distribution of non-trivial zeros of Dirichlet L-functions and the Riemann zeta function. The research focus is on further illuminating the influence of hypothetical zeros of the L-functions having real part different from 1/2. A second project deals with the limitations of sieve methods, which are powerful techniques for bounding the number of primes (or numbers with a bounded number of prime factors) in an integer sequence. The goal of the proposer's work is an improved understanding of how the distribution of primes in a sequence is influenced by the distribution of the sequence in arithmetic progressions and the distribution of the sequence on numbers with an odd number (respectively and even number) of prime factors. The final project is to improve bounds for the density of integers possessing a divisor in a given interval, a problem which is central to the theory of the distribution of divisors of integers.
The study of prime numbers is more than 2000 years old, and the most important problems revolve around how the primes are distributed. Besides applications to many other branches of mathematics, prime number theory is crucial to modern coding theory and cryptography, e.g. secure Internet communication. The proposed research deals with questions concerning how the primes are distributed within certain sets of integers, and questions on the related subject of the statistical behavior of the distribution of divisors of numbers. The proposer will involve graduate students, post-docs and/or talented undergraduate students (REU) in these research projects.
