On the error term in the prime geodesic theorem
Let \(M\) be a \((m+1)\)-dimensional real hyperbolic complete manifold with finite volume. Let \(\pi_M(x)\) the number of prime closed geodesics with length at most \(\log x\). For \(k=0,\dots,m+1\), let \(\Delta_k\) be the Hodge Laplacian acting on the space \(\Omega^k\) of \(k\)-differential forms : the spectral decomposition of \(\Omega^k\) is related to some principal series spaces \(\pi_{\pm,\lambda}\) with \(\lambda\in\Lambda(k)\) a sequence related to the spectrum of \(\Delta_k\).NEWLINENEWLINEThe authors prove an asymptotic expansion with remainder for the counting function \(\pi_M\), where the significant terms are related to the small eigenvalues of the Hodge Laplacians: NEWLINE\[NEWLINE\pi_M(x)=\sum_{\frac34m< s(k)\leq m}(-1)^k\text{li}(x^{s(k)})+{\mathcal O} (x^{\frac34m}(\log x)^{-1}) NEWLINE\]NEWLINE when \(x\to+\infty\). The sum includes, for any integers \(k\) between \(0\) and \(m+1\), all \(s(k)\) such that \((s(k)-k)(m-s(k)-k)\) is a small eigenvalue in \([0,3m^2/16]\) for the Hodge Laplacian \(\Delta_k\), this eigenspace being associated to the principal series \(\pi_{\pm,\lambda(k)}\) such that \(s(k)=\frac m2+\varepsilon\text{i}\lambda(k)\in(\frac32,m]\) with \(\varepsilon=1\) or \(\varepsilon=-1\).NEWLINENEWLINEThis result is a refinement of the asymptotic expansion obtained by \textit{J. Park} [in: Casimir force, Casimir operators and Riemann hypothesis. Mathematics for innovation in industry and science. Proceedings of the conference, Fukuoka, Japan, 2009. Berlin: de Gruyter, 89--104 (2010; Zbl 1290.11124)] as a consequence of the existence of a meromorphic extension of order \(m+1\) for the Ruelle zeta function attached to the manifold \(M\). These asymptotics are optimal in the case of hyperbolic surfaces.
scientific article; zbMATH DE number 6023966