The zero-free region of the derivative of Selberg zeta functions
In the paper under review, the author proves zero-free regions for the derivative of Selberg zeta functions for compact two and three-dimensional hyperbolic manifolds. To be precise, the author proves the following two theorems (with the numbering given in the paper): Theorem 2. Let \(\Gamma\) be a cocompact Fuchsian group with \(\Gamma\backslash\mathbb{H}^{2}\) a Riemann surface of genus \(g\geq2\) and \(Z_{\Gamma\backslash\mathbb{H}^{2}}(s)\) the Selberg zeta function of the corresponding Riemann surface. Then \(Z_{\Gamma\backslash\mathbb{H}}'(s)\neq0\) for \(\text{Re}\, s<\frac{1}{2}\) and \(\text{Im}\, s\neq0\). Theorem 3. Let \(\Gamma\) be a cocompact discrete subgroup of \(\text{PSL}_{2}\left(\mathbb{C}\right)\) and \(Z_{\Gamma\backslash\mathbb{H}^{3}}(s)\) the Selberg zeta function of the corresponding three-dimensional Riemannian space. Then \(Z_{\Gamma\backslash\mathbb{H}}'(s)\neq 0\) for \(\text{Re}\,s<1\) and \(\text{Im}\,s\neq 0\). These two theorems generalize a result of \textit{W. Luo} [Am. J. Math. 127, No. 5, 1141--1151 (2005; Zbl 1109.11040)]. The proofs essentially use only the functional equation and Hadamard product expansions of the completed functions. The theorems cited above were previously used by the author in [Ill. J. Math. 52, No.~4, 1165--1182 (2008; Zbl 1204.11148)] and the review of this item also contains a brief discussion of possible consequences and applications of these theorems.
scientific article; zbMATH DE number 5708705
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