Communications in Mathematical Physics

, Volume 154, Issue 3, pp 433–469 | Cite as

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

  • Pavel M. Bleher
  • Zheming Cheng
  • Freeman J. Dyson
  • Joel L. Lebowitz


We investigate the fluctuations inN α (R), the number of lattice pointsnZ2 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≦RT, 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(−x4+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 ofDα are infinite at every rational point α∈Q2, soDα is analytic nowhere.


Neural Network Distribution Function Statistical Physic Complex System Error Term 
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Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Pavel M. Bleher
    • 1
  • Zheming Cheng
    • 2
  • Freeman J. Dyson
    • 1
  • Joel L. Lebowitz
    • 2
    • 3
  1. 1.School of Natural SciencesInstitute for Advanced StudyPrincetonUSA
  2. 2.Department of MathematicsRutgers UniversityNew BrunswickUSA
  3. 3.Department of PhysicsRutgers UniversityNew BrunswickUSA

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