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Distribution of the error term for the number of lattice points inside a shifted circle
| Content Provider | Semantic Scholar |
|---|---|
| Author | Bleher, Pavel Cheng, Zheming Dyson, Freeman J. Lebowitz, Joel L. |
| Copyright Year | 1993 |
| Abstract | 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. |
| Starting Page | 433 |
| Ending Page | 469 |
| Page Count | 37 |
| File Format | PDF HTM / HTML |
| DOI | 10.1007/BF02102104 |
| Volume Number | 154 |
| Alternate Webpage(s) | http://cmsr.rutgers.edu/images/people/lebowitz_joel/publications/jll.pub_347.pdf |
| Alternate Webpage(s) | http://www.math.iupui.edu/~bleher/Papers/1993_Distribution_Error_Term.pdf |
| Alternate Webpage(s) | https://www.math.iupui.edu/~bleher/Papers/1993_Distribution_Error_Term.pdf |
| Alternate Webpage(s) | https://doi.org/10.1007/BF02102104 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
