Kloosterman sums and the divisor function

Description:

The first project aims to give applications of Kloosterman sums to the divisor function with a goal of breaking through Selberg’s 2/3-barrier with a minimal amount of averaging. The student will work on applications to other arithmetic functions, e.g. to multiple divisor function and the distribution of square-free numbers in progressions, with the same general goal: breaking the existing barrier by introducing as little amount of averaging as possible, for example over residue classes and/or moduli. In particular, we aim to use modern developments in the area of bounding bilinear forms with Kloosterman sums to improve previous results. Moreover, there exists a gap in known bounds for relevant exponential sums with arbitrary composite modulus compared with squarefree. By developing techniques for counting singular points of bounded height on varieties, we expect to make progress towards closing this gap.