Keller, H. B.
(1997)
*Numerical Studies of the Gauss Lattice Problem.*
CRPC,
97-1.
California Institute of Technology
, Pasadena, CA.
https://resolver.caltech.edu/CaltechAUTHORS:20091022-102132378

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## Abstract

The difference between the number of lattice points N(R) that lie in x^2 + y^2 ≤ R^2 and the area of that circle, d(R) = N(R) - πR^2, can be bounded by |d(R)| ≤ KR^θ. Gauss showed that this holds for θ = 1, but the least value for which it holds is an open problem in number theory. We have sought numerical evidence by tabulating N(R) up to R ≈ 55,000. From the convex hull bounding log |d(R)| versus log R we obtain the bound θ ≤ 0.575, which is significantly better than the best analytical result θ ≤ 0.6301 ... due to Huxley. The behavior of d(R) is of interest to those studying quantum chaos.

Item Type: | Report or Paper (Technical Report) | ||||
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Series Name: | CRPC | ||||

Issue or Number: | 97-1 | ||||

Record Number: | CaltechAUTHORS:20091022-102132378 | ||||

Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20091022-102132378 | ||||

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ID Code: | 16440 | ||||

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Deposited On: | 27 Oct 2009 18:45 | ||||

Last Modified: | 03 Oct 2019 01:11 |

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