Arithmetic, geometry and dynamics in the unit tangent bundle of the modular orbifold

Inspired by the work of Zagier, we study geometrically the probability measures $m_y$ with support on the closed horocycles of the unit tangent bundle $M=text{PSL}(2,mathbb{R})/text{PSL}(2,mathbb{Z})$ of the modular orbifold $text{PSL}(2,mathbb Z)$. In fact, the canonical projection $mathfrak{p}:Mtomathbb{H}/text{PSL}(2,mathbb Z)$ it is actually a Seifert fibration over the orbifold with two especial circle fibers corresponding to the two conical points of the modular orbifold. Zagier proved that $m_y$ converges to normalized Haar measure $m_o$ of $M$ as $yto0$: for every smooth function $f:Mto mathbb R$ with compact support $m_y(f)=m_0(f)+o(y^frac12)$ as $yto0$. He also shows that $m_y(f)=m_0(f)+o(y^{frac34-epsilon})$ for all $epsilon>0$ and smooth function $f$ with compact support in $M$ if and only if the Riemann hypothesis is true. In this paper we show that the exponent $frac12$ is optimal if $f$ is the characteristic function of certain open sets in $M$. This of course does not imply that the Riemann hypothesis is false. It is required the differentiability of the functions in the theorem.