Giacomo Cherubini — Page 2

arXiv2018

Prime Geodesic Theorem in the 3-dimensional Hyperbolic Space

For $Γ$ a cofinite Kleinian group acting on $\mathbb{H}^3$, we study the Prime Geodesic Theorem on $M=Γ\backslash \mathbb{H}^3$, which asks about the asymptotic behaviour of lengths of primitive closed geodesics (prime geodesics) on $M$. Let $E_Γ(X)$ be the error in the counting of prime geodesics with length at most $\log X$. For the Picard manifold, $Γ=\mathrm{PSL}(2,\mathbb{Z}[i])$, we improve the classical bound of Sarnak, $E_Γ(X)=O(X^{5/3+ε})$, to $E_Γ(X)=O(X^{13/8+ε})$. In the process we obtain a mean subconvexity estimate for the Rankin-Selberg $L$-function attached to Maass-Hecke cusp forms. We also investigate the second moment of $E_Γ(X)$ for a general cofinite group $Γ$, and show that it is bounded by $O(X^{16/5+ε})$.

math.NT↗

arXiv2015

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 $e_α(s)$ the fractional integral to order $α$ of $e(s)$. We prove that for any small $α>0$ the asymptotic variance of $e_α(s)$ is finite, and given by an explicit expression. Moreover, we prove that $e_α(s)$ has a limiting distribution.

math.NT↗