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 + 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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