Is there a heuristic argument behind the exponent in the [circle problem](https://en.wikipedia.org/wiki/Gauss_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.



Heuristics behind the Circle problem?
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.




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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 $\sqrt M\propto\sqrt R$. 


![](https://ilorentz.org/beenakker/MO/Gauss_circle.png)


Source: [The Circle Problem of Gauss and the Divisor Problem of Dirichlet—Still Unsolved](https://faculty.math.illinois.edu/~berndt/articles/circledivisorsurvey.pdf).  

 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. 


