We study the mean square of the error term in the Gauss circle problem. A heuristic argument based on the consideration of off-diagonal terms in the mean square of the relevant Voronoi-type summation formula leads to a precise conjecture for the mean square of this discrepancy.
"The mean square discrepancy in the circle problem." Mosc. J. Comb. Number Theory 8 (3) 263 - 287, 2019. https://doi.org/10.2140/moscow.2019.8.263