The theory of numbers is a very active research area in mathematics, with connections across the entire discipline and with wide-reaching consequences for digital security. This proposal principally concerns analytic number theory, which is that aspect of the subject that aims to answer quantitative questions. Examples include: What does a typical integer look like 'statistically', in terms of the number and size of its prime factors? How many primes are there up to a given height? What about primes differing from another prime by 2 (so called twin primes)? This proposal considers a number of questions of this kind, not only for natural numbers but also for other number systems that have proven arithmetically significant (corresponding to number fields and function fields).
Three specific topics are considered: The first concerns splitting statistics of primes in algebraic number fields. Elliott and Linnik--Vinogradov developed a method for bounding the least rational prime with a given splitting type in an abelian extension of Q. With Micah Milinovich, Pollack will investigate extending their ideas to certain nonabelian extensions. The second topic is the theory of arithmetic functions, especially as it connects with probabilistic number theory. One problem to be considered is that of obtaining error bounds in a classical theorem of Erdos and Wintner. Finally, Pollack will continue his studies of the distribution of irreducible polynomials over finite fields, with particular attention paid to problems motivated by rational prime number theory. For instance, Pollack will study special configurations of irreducible polynomials, including analogues of the prime k-tuples conjecture and the Bateman--Horn conjecture.
