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Inspired by the work of Zagier, we study geometrically the probabilitymeasures $m_y$ 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$\mathfrak{p}:M\to\mathbb{H}/\text{PSL}(2,\mathbb Z)$ it is actually a Seifertfibration over the orbifold with two especial circle fibers corresponding tothe two conical points of the modular orbifold. Zagier proved that $m_y$converges to normalized Haar measure $m_o$ of $M$ as $y\to0$: for every smoothfunction $f:M\to \mathbb R$ with compact support $m_y(f)=m_0(f)+o(y^\frac12)$as $y\to0$. He also shows that  $m_y(f)=m_0(f)+o(y^{\frac34-\epsilon})$ for all $\epsilon>0$ and smoothfunction $f$ with compact support in $M$ if and only if the Riemann hypothesisis true. In this paper we show that the exponent $\frac12$ is optimal if $f$ isthe characteristic function of certain open sets in $M$. This of course doesnot imply that the Riemann hypothesis is false. It is required thedifferentiability of the functions in the theorem.

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