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

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