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

### Giacomo Cherubini

University of Copenhagen, Denmark### Morten S. Risager

University of Copenhagen, Denmark

## Abstract

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.

## Cite this article

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. 655–685

DOI 10.4171/RMI/1000