Prime Geodesic Theorems for Compact Locally Symmetric Spaces of Real Rank One

Dženan Gušić

doi:10.3390/math8101762

Abstract

Our basic objects will be compact, even-dimensional, locally symmetric Riemannian manifolds with strictly negative sectional curvature. The goal of the present paper is to investigate the prime geodesic theorems that are associated with this class of spaces. First, following classical Randol’s appraoch in the compact Riemann surface case, we improve the error term in the corresponding result. Second, we reduce the exponent in the newly acquired remainder by using the Gallagher–Koyama techniques. In particular, we improve DeGeorge’s bound Oxη, 2ρ − ρn ≤ η < 2ρ up to Ox2ρ−ρnlogx−1, and reduce the exponent 2ρ − ρn replacing it by 2ρ − ρ4n+14n2+1 outside a set of finite logarithmic measure. As usual, n denotes the dimension of the underlying locally symmetric space, and ρ is the half-sum of the positive roots. The obtained prime geodesic theorem coincides with the best known results proved for compact Riemann surfaces, hyperbolic three-manifolds, and real hyperbolic manifolds with cusps.

Highlights

We assume that the Riemannian metric over Y induced from the Killing form is normalized, so that the sectional curvature of Y varies between −4 and −1
The motivation to work within the described setting, i.e., with compact, even-dimensional, locally symmetric Riemannian manifolds of strictly negative sectional curvature, stems from the ρ author’s desire to improve the best known error term O x2ρ− n in the corresponding prime geodesic ρ theorem (1), dating back to 1977 up to O x2ρ− n−1, and the wish to further reduce the ρ ρ exponent 2ρ − n of x in the Gallagherian sense, which is, the wish to replace 2ρ − n with better, 4n+1 smaller one 2ρ − ρ 4n
Where N (t) = Atn + O tn−1 is the number of singularities of the Selberg zeta function ZS (s, τ ) at points i x, 0 < x < t, and A is some explicitly known constant (see, [40] (p. 89, Th. 9.1))

Summary

Introduction

The motivation to work within the described setting, i.e., with compact, even-dimensional, locally symmetric Riemannian manifolds of strictly negative sectional curvature, stems from the ρ author’s desire to improve the best known error term O x2ρ− n in the corresponding prime geodesic ρ theorem (1), dating back to 1977 up to O x2ρ− n (log x )−1 , and the wish to further reduce the ρ ρ exponent 2ρ − n of x in the Gallagherian sense, which is, the wish to replace 2ρ − n with better, 4n+1 smaller one 2ρ − ρ 4n. In the same way Theorem 1 in [6] (see, pages 367 and 371) represents the answer to this query in the case of real hyperbolic manifolds with cusps, Theorems 1 and 2 can be interpreted as the answer to the same query in the case at hand

Preliminary Material

Prime Geodesic Theorem

Gallagherian Prime Geodesic Theorem

Discussion