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

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.