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Numerical Studies of the Gauss Lattice Problem

Keller, H. B. (1997) Numerical Studies of the Gauss Lattice Problem. CRPC, 97-1. California Institute of Technology , Pasadena, CA. http://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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Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20091022-102132378
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ID Code:16440
Collection:CaltechAUTHORS
Deposited By: Kristin Buxton
Deposited On:27 Oct 2009 18:45
Last Modified:26 Dec 2012 11:30

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