6 0. Harmonic Analysis on the Euclidean Plane which usually constitutes the main term. Taking a suitable kernel (a smooth approximation to a step function) and using standard estimates for Bessel’s function, we derive the formula ≤x r( ) = πx + O(x1/3), which was originally established by Voronoi and Sierpinski by different means. The left side counts integral points in the circle of radius √ x. This is also equal to the number of eigenvalues λ(ϕ) ≤ 4π2x (counted with mul- tiplicity), so we have # ϕ : λ(ϕ) ≤ T = T 4π + O(T 1/3 ). In view of the above connection the Gauss circle problem becomes the Weyl law for the operator D (see Section 11.1).

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