Prime scattering geodesic theorem

TL;DR

This work studies scattering geodesics on the modular surface $\mathcal{M}=\mathbb{H}/\mathrm{PSL}(2,\mathbb{Z})$, introducing a finite sojourn-time and proving a prime-number-type theorem for the count of such geodesics with bounded sojourn time. It establishes a precise one-to-one correspondence between scattering geodesics and a subset $\mathcal{G}$ of rationals in $[0,1)$, derived from arithmetic conditions on denominators $q$ and numerators $p$, with the sojourn time given by $l_{\gamma}=2\log(qT_0)$. The main result yields $\Pi(Y)=\frac{3Y}{2\pi^2T_0^2}+O_{T_0}(\sqrt{Y}(\log Y)^{2/3}(\log\log Y)^{1/3})$, linking geodesic counting to number-theoretic objects; further arithmetic preliminaries provide asymptotics for sums $S(x)$ and $\Psi(x)$ using Dirichlet $L$-functions and Wiener–Ikehara theory. Overall, the paper connects the geometry of noncompact hyperbolic surfaces with multiplicative number theory, extending prime-geodesic-type counting to scattering geodesics on the modular surface.

Abstract

The modular surface, given by the quotient $\mathcal{M} = \Ha/\text{PSL}(2,\Z)$, can be partitioned into a compact subset $\Mm$ and an open neighborhood of the unique cusp in $\mathcal{M}$. We consider scattering geodesics in $\mathcal{M}$, first introduced by Victor Guillemin in \cite{Guillemin1976-xr} for hyperbolic surfaces with cusps. These are geodesics in $\mathcal{M}$ that lie in $\mathcal{M} \setminus \Mm$ for both large positive and negative times. Associated with such a scattering geodesic in $\mathcal{M}$, a finite \textit{sojourn time} is defined in \cite{Guillemin1976-xr}. In this article, we study the distribution of these scattering geodesics in $\mathcal{M}$ and their associated \textit{sojourn times}. In this process, we establish a connection between the counting of scattering geodesics on the modular surface and the study of positive integers whose prime divisors lie in arithmetic progression. This article is the first such result for scattering geodesics.

Figures (4)

• Figure 1: Standard cusp neighbourhood in the $\mathbb{H}$. (In this figure, the lines $x = 1/2$ and $x = -1/2$ are identified.)
• Figure 2: Fundamental domain for the PSL(2,$\mathbb{Z}$) action on $\mathbb{H}$.
• Figure 3: Density Histogram of first 10,000 elements in $\mathcal{G}$.
• Figure 4: $\bar{\gamma}_w$:= A lift in $\mathbb{H}$ of a scattering geodesic $\gamma$ in $\mathcal{M}$.

Theorems & Definitions (31)

• Theorem 1.1: Guillemin
• Theorem 1.2
• Remark 1.3
• Definition 1.1
• Theorem 1.4
• Theorem 1.5
• Definition 1.2
• Remark 1.6
• Definition 1.3
• Remark 1.7