Prime geodesics and averages of the Zagier L-series

@article{Balkanova2019PrimeGA,
  title={Prime geodesics and averages of the Zagier L-series},
  author={Olga Balkanova and Dmitry Frolenkov and Morten Skarsholm Risager},
  journal={Mathematical Proceedings of the Cambridge Philosophical Society},
  year={2019}
}
The Zagier L-series encode data of real quadratic fields. We study the average size of these L-series, and prove asymptotic expansions and omega results for the expansion. We then show how the error term in the asymptotic expansion can be used to obtain error terms in the prime geodesic theorem. 

On von Koch Theorem for PSL(2,$$\mathbb {Z}$$)

Under a previously studied condition on the argument of the Selberg zeta function on the critical line, we reach the critical exponent $$\frac{1}{2}$$ in the error term of the prime geodesic

The prime geodesic theorem for $\mathrm{PSL}_{2}(\mathbb{Z}[i])$ 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

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