Optimal spectral rectangles and lattice ellipses

We consider the problem of minimizing the kth eigenvalue of rectangles with unit area and Dirichlet boundary conditions. This problem corresponds to finding the ellipse centred at the origin with axes on the horizontal and vertical axes with the smallest area containing k integer lattice points in the first quadrant. We show that, as k goes to infinity, the optimal rectangle approaches the square and, correspondingly, the optimal ellipse approaches the circle. We also provide a computational method for determining optimal rectangles for any k and relate the rate of convergence to the square with the conjectured error term for Gauss's circle problem.