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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
Article

Abstract

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.

Keywords

Neural Network Distribution Function Statistical Physic Complex System Error Term 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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