| Summary: | 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 <inline-formula><math display="inline"><semantics><mrow><mi>O</mi><mfenced separators="" open="(" close=")"><msup><mi>x</mi><mi>η</mi></msup></mfenced></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><mn>2</mn><mi>ρ</mi></mrow></semantics></math></inline-formula> − <inline-formula><math display="inline"><semantics><mfrac><mi>ρ</mi><mi>n</mi></mfrac></semantics></math></inline-formula> ≤ <inline-formula><math display="inline"><semantics><mi>η</mi></semantics></math></inline-formula> < <inline-formula><math display="inline"><semantics><mrow><mn>2</mn><mi>ρ</mi></mrow></semantics></math></inline-formula> up to <inline-formula><math display="inline"><semantics><mrow><mi>O</mi><mfenced separators="" open="(" close=")"><msup><mi>x</mi><mrow><mn>2</mn><mi>ρ</mi><mo>−</mo><mfrac><mi>ρ</mi><mi>η</mi></mfrac></mrow></msup><msup><mfenced separators="" open="(" close=")"><mo form="prefix">log</mo><mi>x</mi></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup></mfenced></mrow></semantics></math></inline-formula>, and reduce the exponent <inline-formula><math display="inline"><semantics><mrow><mn>2</mn><mi>ρ</mi></mrow></semantics></math></inline-formula> − <inline-formula><math display="inline"><semantics><mfrac><mi>ρ</mi><mi>n</mi></mfrac></semantics></math></inline-formula> replacing it by <inline-formula><math display="inline"><semantics><mrow><mn>2</mn><mi>ρ</mi></mrow></semantics></math></inline-formula> − <inline-formula><math display="inline"><semantics><mrow><mi>ρ</mi><mfrac><mrow><mn>4</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>4</mn><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></semantics></math></inline-formula> outside a set of finite logarithmic measure. As usual, <i>n</i> denotes the dimension of the underlying locally symmetric space, and <inline-formula><math display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula> 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.
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