Prime geodesic theorem for the Picard manifold under the mean-Lindelöf hypothesis. (Q2753165)

Let \(\Gamma=\text{PSL}(2,{\mathcal O})\) act on the upper half space smodel of three-dimensional hyperbolic space where \(\mathcal O\) denotes the ring of integers of an imaginary quadratic number field, and let \(\pi_\Gamma(X)\) denote the number of primitive hyperbolic or loxodromic conjugacy classes of \(\Gamma\) of norm at most \(X\). Then the so-called prime geodesic theorem asserts that NEWLINE\[NEWLINE\pi_\Gamma(X)\sim \text{li}\,X^2\text{ for }X\to\infty.NEWLINE\]NEWLINE If \(\mathcal O\) has class-number one the best error term in this asymptotic law known so far is \(O(X^{(5/3)+ \varepsilon})\). The main result of the paper under review gives a conditional improvement on this error term for the Picard group, namely: Under the assumption of the mean version in the \(\lambda\)-aspect of the Lindelöf hypothesis for the second symmetric power \(L\)-function we have for the group \(\Gamma=\text{PSL}(2,\mathbb{Z}[i])\) and any \(\varepsilon>0\) NEWLINE\[NEWLINE\pi_\Gamma(X)=\text{li}\,X^2+O(X^{(11/7)+ \varepsilon})\text{ as }X\to\infty.NEWLINE\]NEWLINE The reason for the restriction to the Picard group is that the Kuznetsov formula has not been worked out explicitly enough for more general cases.