Prime Geodesic Theorem for Arithmetic Compact Surfaces: Principal Congruence Case
Abstract
We generalize Koyama's 7/10 bound of the error term in the prime geodesic theorems to the principal congruence subgroups for quaternion algebras. Our method avoids the spectral side of the Jacquet--Langlands correspondences, and relates the counting function directly to those for the principal congruence subgroups of Eichler orders of level less than one.
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