DISTRIBUTION OF THE ERROR TERM FOR THE NUMBER OF LATTICE POINTS INSIDE A SHIFTED CIRCLE

We investigate the fluctuations in N(alpha)(R), the number of lattice points n is-an-element-of Z2 inside a circle of radius R centered at a fixed point alpha is-an-element-of [0, 1)2 . Assuming that R is smoothly (e.g., uniformly) distributed on a segment 0 less-than-or-equal-to R less-than-or-equal-to T, we prove that the random variable N(alpha)(R) - piR2/square-root R has a limit distribution as T --> infinity (independent of the distribution of R), which is absolutely continuous with respect to Lebesgue measure. The density p(alpha)(x) is an entire function of x which decays, for real x, faster than exp(- Absolute value of x4-epsilon). We also obtain a lower bound on the distribution function P(alpha)(x) = integral-x/-infinity p(alpha)(y)dy which shows that P(alpha)(- x) and 1 - P(alpha)(x) decay when x --> infinity not faster than exp(- x4+epsilon). Numerical studies show that the profile of the density p(alpha)(x) can be very different for different alpha. For instance, it can be both unimodal and bimodal. We show that integral-infinity/-infinity xp(alpha)(x)dx = 0, and the variance D(alpha) = integral-infinity/-infinity x2p(alpha)(x)dx depends continuously on alpha. However, the partial derivatives of D(alpha) are infinite at every rational point alpha is-an-element-of Q2, so D(alpha) is analytic nowhere.