# The Prime Geodesic Theorem for $\mathrm{PSL}_{2}(\mathbb{Z}[i])$ and Spectral Exponential Sums

## TL;DR

This paper proves new bounds for the error term in the Prime Geodesic Theorem for the Picard manifold, using advanced spectral and exponential sum analysis, and supports the conjecture of a smaller error term through numerical experiments.

## Contribution

It establishes average and pointwise bounds for the error term in the Prime Geodesic Theorem for $ ext{PSL}_2(bZ[i])$, including a spectral exponential sum asymptotic and novel analysis of Kloosterman sums.

## Key findings

- Error term $E_{ ext{ ext{Gamma}}}(X)$ is $O_{ ext{ ext{epsilon}}}(X^{3/2+ ext{epsilon}})$ on average.
- Numerical experiments suggest $E_{ ext{ ext{Gamma}}}(X)$ obeys a conjectural bound of $O_{ ext{ ext{epsilon}}}(X^{1+ ext{epsilon}})$.
- Analysis involves sums of Kloosterman sums and Weyl-strength subconvexity for quadratic Dirichlet $L$-functions.

## Abstract

This work addresses the Prime Geodesic Theorem for the Picard manifold $\mathcal{M} = \mathrm{PSL}_{2}(\mathbb{Z}[i]) \backslash \mathfrak{h}^{3}$, which asks for the asymptotic evaluation of a counting function for the closed geodesics on $\mathcal{M}$. Let $E_{\Gamma}(X)$ be the error term in the Prime Geodesic Theorem. We establish that $E_{\Gamma}(X) = O_{\varepsilon}(X^{3/2+\varepsilon})$ on average as well as many pointwise bounds. The second moment bound parallels an analogous result for $\Gamma = \mathrm{PSL}_{2}(\mathbb{Z})$ 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 $\mathbb{Q}(i)$. 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 $E_{\Gamma}(X)$ obeys a conjectural bound of size $O_{\epsilon}(X^{1+\varepsilon})$.