Sacha Mangerel, University of Montreal
An equivalent form of the famous Erdos Discrepancy Problem, proved by Tao building on the work of the Polymath5 project, states that any completely multiplicative function taking values on the unit circle has unbounded partial sums. It was observed in the course of the Polymath5 project that the same is not true if one considers the most natural translation of this problem to the ring F_q[t] of polynomials over a finite field.
We will discuss recent joint work with O. Klurman and J. Teräväinen demonstrating that the function field discrepancy problem depends heavily on the way the elements of the sums are ordered, in contrast to the integer setting. In particular, we will introduce three different notions of discrepancy, and discuss the problem of classifying those completely multiplicative functions that have uniformly bounded partial sums with respect to each of these notions. We will also address the problem of bounding the minimal rate of growth of unbounded partial sums, which is the subject of some speculation in the integer setting.