I recently gave a talk at the Dartmouth Number Theory Seminar (thank you Edgar for inviting me and to Edgar, Naomi, and John for being such good hosts). In this talk, I described the recent successes we've had on working with variants of the Gauss Circle Problem.
The story began when (with Tom Hulse, Chan Ieong Kuan, and Alex Walker — and with helpful input from Mehmet Kiral, Jeff Hoffstein, and others) we introduced and studied the Dirichlet series $$\begin{equation} \sum_{n \geq 1} \frac{S(n)^2}{n^s}, \notag \end{equation}$$ where $S(n)$ is a sum of the first $n$ Fourier coefficients of an automorphic form on GL(2). We've done this successfully with a variety of automorphic forms, leading to new results for averages, short-interval averages, sign changes, and mean-square estimates of the error for several classical problems. Many of these papers and results have been discussed in other places on this site.
Ultimately, the problem becomes acquiring sufficiently detailed understandings of the spectral behavior of various forms (or more correctly, the behavior of the spectral expansion of a Poincare series against various forms). We are continuing to research and study a variety of problems through this general approach.
The slides for this talk are available here.
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