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 $M=\text{PSL}(2,\mathbb{R})\mathrm{/}\text{PSL}(2,\mathbb{Z})$ of the modular orbifold $\text{PSL}(2,\mathbb{Z})$. In fact, the canonical projection $\mathfrak{p}:M\to \mathbb{H}\mathrm{/}\text{PSL}(2,\mathbb{Z})$ 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 $y\to 0$: for every smooth function $f:M\to \mathbb{R}$ with compact support $m_y(f)=m_0(f)+o(y^{\frac{1}{2}})$ as $y\to 0$. He also shows that $m_y(f)=m_0(f)+o(y^{\frac{3}{4}-ϵ})$ for all $ϵ>0$ 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 $\frac{1}{2}$ 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.