Arithmetic forms of Selberg zeta functions with applications to prime geodesic theorem
Excerpts from portions of the introduction: ``Let \(\Gamma\) be a discrete subgroup of \(\text{SL}_2(\mathbb{R})\) containing \(-1_2\) with finite covolume \(v(\Gamma\setminus{\mathfrak H})\), \({\mathfrak H}\) denoting the upper half plane. The Selberg zeta-function attached to \(\Gamma\) is defined by \[ Z_\Gamma(s):= \prod_{\{P\}_\Gamma} \prod_{m=0}^\infty (1-N(P)^{-s-m}), \quad(\operatorname {Re}(s)>1), \] where \(\{P\}_\Gamma\) runs through all primitive hyperbolic conjugacy classes of \(\Gamma\) with \(\text{tr}(P)>2\), and \(N(P):= |\rho|^2\) with \(\rho\) the eigenvalue of \(P\in \Gamma\) such that\(|\rho|> 1\). \dots{} For any basis \(\{u_i\}\) of \({\mathcal O}\) [a maximal order of \(B\), an indefinite division quaternion algebra over \(\mathbb{Q}\)] over \(\mathbb{Z}\), set \[ d(B)= |\det (\operatorname {tr}(u_iu_j))|^{1/2}. \] Put \[ {\mathcal D}:= \{D\in \mathbb{Z}_{>0}\mid D\equiv 0,1\pmod 4,\text{ not a square}\}. \] Let \({\mathfrak o}\) be an order of \(K= \mathbb{Q}(\sqrt{D})\) and \(h({\mathfrak o})= h(D)\) be the number of classes of proper \({\mathfrak o}\)-ideals in the narrow sense. We moreover set \[ \lambda(K)= \prod_{p\mid d(B)} \Bigl(1 -\bigl(\tfrac Kp\bigr)\Bigr), \] where \((K/p)\) denotes the Artin symbol for \(K= \mathbb{Q}(\sqrt{D})\). Let \(\varepsilon_D= (\alpha+ \beta\sqrt{D})/2\) with \((\alpha,\beta)\) being the minimal solution of the Pell equation: \(x^2- Dy^2=4\). The main theorem of this paper is as follows. Theorem 1.1. Let \(B\) be a division indefinite quaternion algebra over \(\mathbb{Q}\). Then \[ Z_B(s)= \mathop{{\prod}^*}_{D>0} \prod_{n=0}^\infty \Bigl(1-\varepsilon_D^{-2(s+n)} \Bigr)^{h(D)\lambda(D)}, \] and \[ \frac{Z_B'}{Z_B}(s)= \mathop{{\sum}^*}_{D>0} \sum_{m=1}^\infty h(D) \lambda(D) \log \varepsilon_D^2\cdot \frac{\varepsilon_D^{-2ms}} {1-\varepsilon_D^{-2m}}, \] where \(\lambda(D)= \lambda(\mathbb{Q} (\sqrt{D}))\) and the symbol * indicates that \(D\) runs through all elements in \({\mathcal D}\) satisfying the following conditions: (Pr-i) \((\frac{K}{p})\neq 1\) for any prime integers \(p\mid d(B)\). (Pr-ii) \((f(D),d(B))= 1\), where the positive integer \(f(D)\) is given by \(D= f(D)^2 D_K\), \(D_K\) being the discriminant of \(K\). Remark. For \(\Gamma= \text{SL}(2,\mathbb{Z})\) and its congruence subgroups. Sarnak obtains such an arithmetic form of \(Z_\Gamma(s)\). Theorem 1.1 has an application of improving the prime geodesic theorem: \[ \pi_\Gamma(x)\sim \text{li}(x)\sim \frac{x}{\log x}, \tag{1.2} \] where \(\pi_\Gamma(x)\) is the number of primitive hyperbolic conjugacy classes \(P\) of \(\Gamma\) whose norm \(N(P)\) satisfies that \(N(P)\leq x\). Theorem 1.4. Let \(B\) be a division indefinite quaternion algebra over \(\mathbb{Q}\). Put \(\pi_B(x)= \pi_{{\mathcal O}^1}(x)\). Then for \(x^{(1/2)} (\log x)^2< y< x\), we have \[ \pi_B(x+y)- \pi_B(x)\ll y. \tag{1.4} \] The implicit constant depends only on \(B\). [Here \({\mathcal O}'\) is a certain sub-order of \({\mathcal O}\).]'' The exponents of \(x+y\) in the range inequality of Theorem 1.4 are best possible.
number of primitive hyperbolic conjugacy classes
scientific article; zbMATH DE number 1902571
scientific article