On the variance of the error term in the hyperbolic circle problem

Cherubini, Giacomo; Risager, Morten S.

doi:10.4171/RMI/1000

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Revista Matemática Iberoamericana

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Volume 34, Issue 2, 2018, pp. 655–685

DOI: 10.4171/RMI/1000

Published online: 2018-05-28

On the variance of the error term in the hyperbolic circle problem

Giacomo Cherubini^[1] and Morten S. Risager^[2]

(1) University of Copenhagen, Denmark
(2) University of Copenhagen, Denmark

Let $e(s)$ be the error term of the hyperbolic circle problem, and denote by $e_\alpha(s)$ the fractional integral to order $\alpha$ of $e(s)$. We prove that for any small $\alpha>0$ the asymptotic variance of $e_\alpha(s)$ is finite, and given by an explicit expression. Moreover, we prove that $e_\alpha(s)$ has a limiting distribution.

Keywords: Hyperbolic lattice points, Selberg’s pre-trace formula, fractional integration

Cherubini Giacomo, Risager Morten: On the variance of the error term in the hyperbolic circle problem. Rev. Mat. Iberoam. 34 (2018), 655-685. doi: 10.4171/RMI/1000