Consider the characteristic function f of a disc of radius r. The classical circle problem asks for a bound on the error term in the expression

$\sum_{(x,y) \in \mathbb{Z}^2} f(x,y) = closed expression + error term$,

where the closed expression happens to be the area of the circle. (The divisor problem is the analogous question for the characteristic function of the domain under a hyperbola $y = r/x$.) Hardy and Landau showed that the error term is no smaller than r^{1/2} or so.

Question: what happens if we let f be a smoothed version of the characteristic function of a disc? Does the error term become conjecturally much smaller (how much smaller?)? What has been proven in that case?

(Obviously the answer will depend on the extent and type of smoothing - how strongly? Would the classical problem be equivalent to a smoothing where the decay happens within an annulus of constant width?)