The prime geodesic theorem for PSL2(Z[i])\mathrm{PSL}_{2}(\mathbb{Z}[i]) and spectral exponential sums

We shall ponder the Prime Geodesic Theorem for the Picard manifold $\mathcal{M} = \mathrm{PSL}_{2}(\mathbb{Z}[i]) \backslash \mathfrak{h}^{3}$, which asks about the asymptotic behaviour of a counting function for the closed geodesics on $\mathcal{M}$. Let $E_{\Gamma}(X)$ be the error term arising from counting prime geodesics, we then prove the bound $E_{\Gamma}(X) \ll X^{3/2+\epsilon}$ on average, as well as various versions of pointwise bounds. The second moment bound is the pure counterpart of work of Balog et al. for $\Gamma = \mathrm{PSL}_{2}(\mathbb{Z})$, and the main innovation entails the delicate analysis of sums of Kloosterman sums with an explicit evaluation of oscillatory integrals. Our pointwise bounds concern Weyl-type subconvex bounds for quadratic Dirichlet $L$-functions over $\mathbb{Q}(i)$. Interestingly, we are also able to establish an asymptotic law for the spectral exponential sum in the spectral aspect for a cofinite Kleinian group $\Gamma$. Finally, we produce numerical experiments of its behaviour, visualising that $E_{\Gamma}(X)$ obeys a conjectural bound of the size $O(X^{1+\epsilon})$.Comment: Numerous improvements to the exposition; improved the quality of the main theorem (Theorem 1.1) and achieved additional theorems such as Theorems 1.4, 3.17, 4.1, and 5.

Prime Geodesic Theorem in the 3-dimensional Hyperbolic Space

For $\Gamma$ a cofinite Kleinian group acting on $\mathbb{H}^3$, we study the Prime Geodesic Theorem on $M=\Gamma \backslash \mathbb{H}^3$, which asks about the asymptotic behaviour of lengths of primitive closed geodesics (prime geodesics) on $M$. Let $E_{\Gamma}(X)$ be the error in the counting of prime geodesics with length at most $\log X$. For the Picard manifold, $\Gamma=\mathrm{PSL}(2,\mathbb{Z}[i])$, we improve the classical bound of Sarnak, $E_{\Gamma}(X)=O(X^{5/3+\epsilon})$, to $E_{\Gamma}(X)=O(X^{13/8+\epsilon})$. In the process we obtain a mean subconvexity estimate for the Rankin-Selberg $L$-function attached to Maass-Hecke cusp forms. We also investigate the second moment of $E_{\Gamma}(X)$ for a general cofinite group $\Gamma$, and show that it is bounded by $O(X^{16/5+\epsilon})$.Comment: Corrected proof of Theorem 3.3 (with a weaker bound), added two authors, 18 page