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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.

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.

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.

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.