# Distribution of the error term for the number of lattice points inside a shifted circle

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## Abstract

We investigate the fluctuations in*N*_{ α }*(R)*, the number of lattice points*n*∈**Z**^{2} inside a circle of radius*R* centered at a fixed point α∈[0, 1)^{2}. Assuming that*R* is smoothly (e.g., uniformly) distributed on a segment 0≦*R*≦*T*, we prove that the random variable\(\frac{{N_\alpha (R) - \pi R^2 }}{{\sqrt R }}\) has a limit distribution as*T*→∞ (independent of the distribution of*R*), which is absolutely continuous with respect to Lebesgue measure. The density*p*_{ α }*(x)* is an entire function of*x* which decays, for real*x*, faster than exp(−|*x*|^{4−ε}). We also obtain a lower bound on the distribution function\(P_\alpha (x) = \int_{ - \infty }^x {p_\alpha (y)} dy\) which shows that*P*_{ α }*(−x)* and 1−*P*_{ α }*(x)* decay when*x*→∞ not faster than exp(−*x*^{4+ε}). Numerical studies show that the profile of the density*p*_{ α }*(x)* can be very different for different α. For instance, it can be both unimodal and bimodal. We show that\(\int_{ - \infty }^\infty {xp_\alpha (x)} dx = 0\), and the variance\(D_\alpha = \int_{ - \infty }^\infty {x^2 p_\alpha (x)} dx\) depends continuously on α. However, the partial derivatives of*D*_{α} are infinite at every rational point α∈**Q**^{2}, so*D*_{α} is analytic nowhere.

## Keywords

Neural Network Distribution Function Statistical Physic Complex System Error Term## Preview

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