Junior Number Theory Seminar

Estimating the correlation $\sum_{n \le x} d_k(n)d(n+h)$ is a central problem in analytic number theory. In this talk, I will present a method to obtain an asymptotic formula for a smoothed version of this sum. A key feature of the result is a power-saving error term whose exponent does not depend on $k$, improving earlier bounds where the quality of the saving deteriorates with $k$. The argument relies on balancing three distinct bounds for the remainder term according to the sizes of the factors of $n$.