Hexagonal Lattice Points on Circles

Details Hexagonal Lattice Points on Circles Hexagonal Lattice Points on Circles

2005-08-11

arxiv.org/abs/math/0508201v1

Mathematics

Number Theory

Master's Thesis. 53 pages, 8 figures

Scientific paper

We study the hexagonal lattice $\mathbb{Z}[\omega]$, where $\omega^6=1$. More specifically, we study the angular distribution of hexagonal lattice points on circles with a fixed radius. We prove that the angles are equidistributed on average, and suggest the possibility of constructing a consistent discrete velocity model (DVM) for the Boltzmann equation, using a hexagonal lattice. Equidistribution on average is expressed in terms of cancellation in exponential sums. We introduce Hecke L-functions and investigate their analytic properties in order to derive estimates on sums of Hecke characters. Using a version of the Halberstam-Richert inequality, these estimates then yield the desired results for the exponential sums. As a further measure of equidistribution, we give a bound for the discrepancy.

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