AUTHORS: Dzenan Gusic
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ABSTRACT: Prime geodesic theorem gives an asymptotic estimate for the number of prime geodesics over underlying symmetric space counted by their lengths. In any setting, the search for the optimal error term is widely open. Our objective is to derive a weighted, generalized form of the prime geodesic theorem for compact, even-dimensional, locally symmetric Riemannian manifolds of strictly negative sectional curvature. We base our methodology on an application of the integrated, Chebyshev-type counting function of appropriate order. The obtained error term improves the corresponding, and best known one in the case of classical prime geodesic theorem. Our conclusion in the case at hand is that a weighted sense yields a better result.
KEYWORDS: Weighted prime geodesic theorem, counting functions, zeta functions, topological singularities, spectral singularities
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