Theory of L-functions, prime numbers and divisors

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

This project is concerned with investigations into the distribution of zeros of L-functions, prime numbers and divisors of integers. The distribution of prime numbers and zeros of L-functions have long been known to be related, but some relations have only recently come to light. Together with A. Zaharescu and K. Soundararajan, the investigator is exploring connections between fractional parts of imaginary parts of zeros of L-functions, counts of prime numbers in short intervals, and the pair correlation of zeros of L-functions. A second area of study involves uncovering subtle inequities in the distribution of numbers with exactly k prime factors (k>1) in arithmetic progressions, problems intimately connected with the location of the zeros of Dirichlet L-functions. A focus of the research is examining differences between the the behavior in arithmetic progressions of primes (k=1) and "almost primes" (k>1). Thirdly, the investigator will continue studying how the divisors of "typical" and "atypical" integers are distributed. Investigations into the probability theory underlying the study of divisors will be a major theme. The positive integers are perhaps the most basic objects in mathematics, and it is important to understand their multiplicative structure - how they may be written as products, both of prime numbers and in general. Questions about the distribution of integers with a certain multiplicative structure, especially prime numbers, are often very difficult and poorly understood. Several problems of this type are being investigated in this project, with particular emphasis on the connections with the behavior of special functions called L-functions and with probability and statistical theory.