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

AbstractWe investigate the fluctuations inNα(R), the number of lattice pointsn∈Z2 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 densitypα(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 thatPα(−x) and 1−Pα(x) decay whenx→∞ not faster than exp(−x4+ε). Numerical studies show that the profile of the densitypα(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 ofDα are infinite at every rational point α∈Q2, soDα is analytic nowhere.