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We determine the variance for the fluctuations of the arithmetic measuresobtained by collecting all closed geodesics on the modular surface with thesame discriminant and ordering them by the latter. This arithmetic variancediffers by subtle factors from the variance that one gets when consideringindividual closed geodesics when ordered by their length. The arithmeticvariance is the same one that appears in the fluctuations of measuresassociated with quantum states on the modular surface.

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