Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
The prime geodesic theorem for $\operatorname{PSL}_{2}(\mathbb{Z}[i])$ and spectral exponential sums

### Ikuya Kaneko

Vol. 16 (2022), No. 8, 1845–1887
##### Abstract

This work addresses the prime geodesic theorem for the Picard manifold $\mathsc{ℳ}={\mathrm{PSL}}_{2}\left(ℤ\left[i\right]\right)\setminus {\mathfrak{𝔥}}^{3}$, which asks for the asymptotic evaluation of a counting function for the closed geodesics on $\mathsc{ℳ}$. Let ${E}_{\Gamma }\left(X\right)$ be the error term in the prime geodesic theorem. We establish that ${E}_{\Gamma }\left(X\right)={O}_{𝜖}\left({X}^{3∕2+𝜖}\right)$ on average as well as many pointwise bounds. The second moment bound parallels an analogous result for $\Gamma ={\mathrm{PSL}}_{2}\left(ℤ\right)$ due to Balog et al. and our innovation features the delicate analysis of sums of Kloosterman sums with an explicit manipulation of oscillatory integrals. The proof of the pointwise bounds requires Weyl-strength subconvexity for quadratic Dirichlet $L$-functions over $ℚ\left(i\right)$. Moreover, an asymptotic formula for a spectral exponential sum in the spectral aspect for a cofinite Kleinian group $\Gamma$ is given. Our numerical experiments visualise in particular that ${E}_{\Gamma }\left(X\right)$ obeys a conjectural bound of size ${O}_{𝜖}\left({X}^{1+𝜖}\right)$.

##### Keywords
prime geodesic theorem, Picard manifold, second moment, $L$-functions, Selberg trace formula, Kuznetsov formula, Kloosterman sums, spectral exponential sums, subconvexity
##### Mathematical Subject Classification
Primary: 11M36
Secondary: 11F72, 11L05, 11M26