The prime geodesic theorem for \(\mathrm{PSL}2(\mathbb{Z}[i])\) and spectral exponential sums
This article deals with the prime geodesic theorem on the Picard manifold \(\Gamma\backslash\mathfrak h^3\) with \(\mathfrak h^3\) the 3-dimensional upper half space and \(\Gamma=\mathrm{PSL}(2,\mathbb Z[\sqrt{-1}])\). Let \[ \Psi_\Gamma(X)=\sum_{N(P)\le X}\log N(P_0), \] where \(P\) runs through hyperbolic and loxodromic conjugacy classes of \(\Gamma\), \(P\) is a power of a primitive class \(P_0\), and \(N(P)\) stands for its norm. The main purpose of this article is to improve the error term \[ E_\Gamma(X)=\Psi_\Gamma(X)-\sum_{1<s_j\le2}\frac{X^{is_j}}{s_j} \] of prime geodesic theorem with \(s_j\) the exceptional zeros of the Selberg zeta function of \(\Gamma\). It is known that subconvexity estimates of automorphic \(L\)-functions play an essential role for improving \(E_\Gamma(X)\). Indeed \textit{S.-y. Koyama} [Forum Math. 13, No. 6, 781--793 (2001; Zbl 1061.11024)] proved that \(E_\Gamma(X)=O_\varepsilon(X^{11/7+\varepsilon})\) under the mean-Lindelöf hypothesis for symmetric square \(L\)-functions associated to the Hecke-Maass forms, which was a conditional improvement of the trivial bound \(O(X^{5/3+\varepsilon})\). In the article under review the author succeeds in obtaining an unconditional bound \[ E_\Gamma(X)=O_\varepsilon(X^{376/237+\varepsilon}) \] as well as in clarifying the relation between the subconvexity estimates and the improvement of \(E_\Gamma(X)\). He also presents a better estimate of \(S(T, X)=\sum_{t_j\le T}X^{it_j}\) with \(s_j=1+it_j\) in the spectral aspect.
scientific article; zbMATH DE number 7628581
scientific article