# Second Moment of the Prime Geodesic Theorem for $\mathrm{PSL}(2, \mathbb{Z}[i])$

## TL;DR

This paper investigates the average behavior of the error term in the Prime Geodesic Theorem for the Picard group, showing that an expected bound holds unconditionally through second moment analysis.

## Contribution

It establishes an unconditional average bound for the error term in the Prime Geodesic Theorem for (2, 1[i]) by analyzing its second moment.

## Key findings

- Unconditional second moment bound for the error term.
- Average error term behaves as predicted under Lindelf6f hypothesis.
- Improves understanding of prime geodesic distribution for (2, 1[i])

## Abstract

The remainder $E_\Gamma(X)$ in the Prime Geodesic Theorem for the Picard group $\Gamma = \mathrm{PSL}(2,\mathbb{Z}[i])$ is known to be bounded by $O(X^{3/2+\epsilon})$ under the assumption of the Lindel\"of hypothesis for quadratic Dirichlet $L$-functions over Gaussian integers. By studying the second moment of $E_\Gamma(X)$, we show that on average the same bound holds unconditionally.