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

Abstract

We investigate the fluctuations inN _α (R), the number of lattice pointsn∈Z ² inside a circle of radiusR centered at a fixed point α∈[0, 1)². Assuming thatR 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 asT→∞ (independent of the distribution ofR), which is absolutely continuous with respect to Lebesgue measure. The densityp _α (x) is an entire function ofx which decays, for realx, 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 thatP _α (−x) and 1−P _α (x) decay whenx→∞ not faster than exp(−x ^4+ε). Numerical studies show that the profile of the densityp _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 ofD _α are infinite at every rational point α∈Q ², soD _α is analytic nowhere.