Number Theory Seminar

Many old questions in analytic number theory are still open. This includes the twin prime conjecture, and its quantitative refinement - the Hardy-Littlewood prime-tuple conjecture. An analogy between number fields and function fields over finite fields, which will be presented at the talk, allows us to ask analogous questions for polynomials defined over finite fields instead of integers. In the function field setting, a new parameter - the cardinality of the finite field itself - enters the picture and allows us to consider the questions from new angles, sometimes shedding light back on the number field setting. We will discuss the previously known results on the twin prime problem in the function field setting and the tools involved. Finally, we will discuss recent joint work with Will Sawin which improves upon some of these results.