Let be the error term of the hyperbolic circle problem, and denote by the fractional integral to order of . We prove that for any small the asymptotic variance of is finite, and given by an explicit expression. Moreover, we prove that has a limiting distribution.
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Giacomo Cherubini, Morten S. Risager, On the variance of the error term in the hyperbolic circle problem. Rev. Mat. Iberoam. 34 (2018), no. 2, pp. 655–685DOI 10.4171/RMI/1000