The variance of the error function in the shifted circle problem is a wild function of the shift
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We prove that the variance of the error function in the shifted circle problem, as a function of the shift, is a continuous function which has a sharp local maximum with infinite derivatives at every rational point on a plane.
KeywordsNeural Network Statistical Physic Continuous Function Complex System Nonlinear Dynamics
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- [B] Bleher, P.M.: On the distribution of the number of lattice points inside a family of convex ovals. Duke Math. J.67, 461–481 (1992)Google Scholar
- [BCDL] Bleher, P.M., Cheng, Zh., Dyson, F.J., Lebowitz, J.L.: Distribution of the error term for the number of lattice points inside a shifted circle. Commun. Math. Phys.154, 433–469 (1993)Google Scholar
- [BD] Bleher, P.M., Dyson, F.J.: Mean square value of exponential sums related to representation of integers as sum of two squares. Preprint IASSNS-HEP-92/84, Institute for Advanced Study, Princeton, 1992Google Scholar
- [C] Cramér, H.: Über zwei Sätze von Herrn G.H. Hardy. Math. Zeit.15, 201–210 (1922)Google Scholar
- [H-B] Heath-Brown, D.R.: The distribution and moments of the error term in the Dirichlet divisor problem. Acta Arithmetica60, 389–415 (1992)Google Scholar
- [K] Kendall, D.G.: On the number of lattice points inside a random oval. Quart. J. Math. (Oxford)19, 1–26 (1948)Google Scholar