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

TL;DR

This paper investigates the error term in the hyperbolic circle problem, demonstrating that its fractional integral has finite asymptotic variance and a limiting distribution, advancing understanding of error behavior in hyperbolic geometry.

Contribution

It establishes the finiteness of the asymptotic variance and the limiting distribution of the fractional integral of the error term in the hyperbolic circle problem.

Findings

Asymptotic variance of the fractional integral is finite.

Explicit expression for the asymptotic variance.

Fractional integral has a limiting distribution.

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.