Spectral estimates for compact hyperbolic space forms and the Selberg zeta function for \(p\)-spectra. II
We maintain the assumptions and the notation of the review of part I of the present work (see Zbl 0868.11047). Let \(S_p\) denote the \(p\)-spectrum of the Laplacian \(\Delta= d\delta +\delta d\) on \(L^2(V)\), and denote by \(d^\delta_p (\mu)\) the dimension of the eigenspace of coclosed \((\delta \alpha= 0)\) eigenforms \(\alpha\) of \(\Delta\) with eigenvalue \(\mu\). Let \(N_p(T)\) denote the sum of the \(d^\delta_p (\mu)\), where the summation extends over all \(\mu\in S_p\) such that \(\mu <T^2+ (p-(n-1)/2)^2\), and let \(c_p: =(4\pi)^{-n/2} {n-1 \choose p} \text{vol} (V)/ \Gamma ({n+2 \over 2})\). Then the main result of the work under review is Theorem B: \(N_p(T)= c_pT^n +O(T^{n-1}/ \log T)\) as \(T\to\infty\). (The reviewer wants to point out that this result is known for \(p=0\) from the work [\textit{J. J. Duistermaat}, \textit{J. A. C. Kolk} and \textit{V. S. Varadarajan}, Spectra of compact locally symmetric manifolds of negative curvature, Invent. Math. 52, 27-93 (1979; Zbl 0434.58019), Erratum ibid. 54, 101 (1979; Zbl 0447.58034); see p. 89, Theorem 9.1], where the gamma factor must be corrected to \(\Gamma^{-1} ({n \over 2} +1)\).) The proof of Theorem B uses the author's Selberg trace formula for the \(p\)-spectrum. This formula enables the author to introduce a suitable Selberg zeta function for the \(p\)-spectrum which in its properties closely resembles the well-known case of the Selberg zeta function for a compact Riemann surface. (The case \(n=3\), \(p=0\) was discussed even for cocompact groups with torsion by \textit{J. Elstrodt}, \textit{F. Grunewald} and \textit{J. Mennicke} [The Selberg zeta-function for cocompact discrete subgroups of \(\text{PSL} (2, \mathbb{C})\). Elementary and analytic theory of numbers, Banach Cent. Publ. 17, 83-120 (1985; Zbl 0599.10015)].) The final proof of Theorem B is based on an application of methods of analytic number theory to the Selberg zeta function for the \(p\)-spectrum. The technical details are much more involved than in the classical case \(n=2\), \(p=0\) considered in \textit{D. A. Hejhal} [The Selberg trace formula for \(\text{PSL} (2,\mathbb{R})\). I, [Lect. Notes Math. 548 (1976; Zbl 0347.10018)].
scientific article; zbMATH DE number 605109