2017/11/09 by Alberto Verjovsky, Verjovsky, Alberto · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1711.03593
openalex publication_date 2017/11/09 · openalex created_date 2017/11/17 · openalex updated_date 2026/07/28
Inspired by the work of Zagier, we study geometrically the probability measures my with support on the closed horocycles of the unit tangent bundle M=PSL(2,ℝ)/PSL(2,ℤ) of the modular orbifold PSL(2,\mathbb Z). In fact, the canonical projection \mathfrakp:M→ℍ/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 my converges to normalized Haar measure mo of M as y→0: for every smooth function f:M→ \mathbb R with compact support my(f)=m0(f)+o(y^\frac12) as y→0. He also shows that my(f)=m0(f)+o(y\frac34-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 \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.