A mean-square bound for the lattice discrepancy of bodies of rotation with flat points on the boundary

where t is a large real parameter. For enlightening accounts on this topic, the reader is referred to E. Krätzel’s monographs [14] and [15], to a recent survey article by A. Ivić, E. Krätzel, M. Kühleitner, and W.G. Nowak [10], and to M. Huxley’s book [7] where he exposed his breakthrough in the planar case (”Discrete Hardy-Littlewood method”). For the case that ∂B is of bounded nonzero Gaussian curvature throughout, the usual and plausible conjecture is that PB(t) ≪ ts (1.2)