The variance of arithmetic measures associated to closed geodesics on the modular surface

Details The variance of arithmetic measures associated to closed geodesics on the modular surface The variance of arithmetic measures associated to closed geodesics on the modular surface

2008-10-18

arxiv.org/abs/0810.3331v2

Mathematics

Number Theory

Added a brief comment concerning Duke's theorem on the unit tangent bundle

Scientific paper

We determine the variance for the fluctuations of the arithmetic measures obtained by collecting all closed geodesics on the modular surface with the same discriminant and ordering them by the latter. This arithmetic variance differs by subtle factors from the variance that one gets when considering individual closed geodesics when ordered by their length. The arithmetic variance is the same one that appears in the fluctuations of measures associated with quantum states on the modular surface.

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