Heuristics behind the Circle problem? Is there a heuristic argument behind the exponent in the circle problem?  The problem that I am referring to is the following:  Consider a circle of radius $R$ centered at the origin in the plane and let $N(R)$ denote the number of integer lattice points contained in the circle.  Then it easy to show that $N(R)/ \pi R^2 \to 1$ as $R \to \infty$.  The circle problem asks what is the optimal exponent for the error term.
 A: The heuristics for this problem is due to Gauss: the error term $\delta N(R)=N(R)-\pi R^2$ must scale with the circumference $2\pi R$ of the circle, because it is along the circumference that the ambiguity of lattice points just inside or just outside the circle appears; Gauss' estimate $\delta N(R)\propto R$ is an overestimate, the configuration  of lattice points is such that the excess and deficit of points just inside and just outside partially cancels each other, so that the exponent is closer to $1/2$. Intuitively, this cancellation is obvious from a figure, but to obtain the correct exponent is not something that I think is accessible by any heuristics.

UPDATE: This could be an intuitive argument ("heuristics") for $\delta N(R)\propto R^{1/2}$: The number $M$ of lattice points along the perimeter is of order $R$. If each lattice point contributes $\pm 1$ to $\delta N(R)$, and these $M$ contributions are statistically independent, the total contribution would be of order $\sqrt M\propto\sqrt R$.  

Source: The Circle Problem of Gauss and the Divisor Problem of Dirichlet—Still Unsolved.
 Each lattice point is associated with a unit cell, chosen such that the lattice point is in the lower-left corner of the cell. Lattice points inside the circle correspond to a cell shown in red. The number of red cells that extends outside the circle cancels approximately with the number of white cells that extend inside, producing a sub-linear scaling of $\delta N$ with the radius $R$ of the circle. 
A: A formula, due to Hardy, expresses the error term $P(x) := N(\sqrt{x})-\pi x$ in terms of values of a Bessel function:
$$(*)\, P(x) = x^{1/2}\sum_{n \ge 1}\frac{r(n)}{n^{1/2}}J_1(2\pi \sqrt{nx}),$$
where $r(n):=\#\{ (a,b)\in \mathbb{Z}^2: a^2+b^2=n\}$ (this should be modified slightly for integer $x$). As $J_1(t)=O(1/\sqrt{t})$ as $t\to \infty$, any truncation of the RHS of $(*)$ is $O(x^{1/4})$. Unfortunately, this does not yield such a bound for $P(x)$, since $\sum r(n)/n^{3/4}$ obviously diverges.
There are also various results studying $P(x)$ in mean, usually using $(*)$ or variations thereof. The first such result seems to be Cramér's, who in 1921 showed that
$$\int_{1}^{x} P^2(t)\, dt \sim C x^{3/2}$$
for an explicit $C>0$. This is one good reason to suspect that $P(t) = O(t^{1/4+\epsilon})$ (and it certainly shows that $P(t) = O(t^{1/4-\epsilon})$ is impossible, which was proven before by Hardy). Subsequent works include computing the 3rd and 4th moments of $P$ (Tsang), and proving that $P(t)/t^{1/4}$ has a limiting distribution (Heath-Brown).
