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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.

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 $2pi R$ 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 $sqrt Mproptosqrt R$.

<sub>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.</sub>

A formula, due to Hardy, expresses the error term $P(x) := N(sqrtx)-pi x$ in terms of values of a Bessel function:
$$(*), P(x) = x^1/2sum_n ge 1fracr(n)n^1/2J_1(2pi sqrtnx),$$
where $r(n):=# (a,b)in mathbbZ^2: a^2+b^2=n$ (this should be modified slightly for integer $x$). As $J_1(t)=O(1/sqrtt)$ as $tto infty$, 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
$$int_1^x P^2(t), dt sim C x^2/3$$
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).