On the structure of the Selberg class. III: Sarnak's rigidity conjecture
In the third paper of this series the authors deal with a problem, raised by Peter Sarnak, about rigidity in the Selberg class \(\mathcal S\) [Part I, cf. Acta Math. 182, 207-241 (1999; Zbl 1126.11335), Part II, J. Reine Angew. Math. 524, 73-96 (2000; Zbl 0982.11051), Part IV, Acta Arith. 104, 97-116 (2002; Zbl 0996.11053)]. Roughly speaking, the rigidity conjecture states that every continuous one-parameter family \({\mathcal F} =\{F(s;\xi)\}_{\xi\in\mathbb R}\) of functions in \(\mathcal S\) is a shifted family, that is, \[ F(s;\xi)=\prod_{j=1}^k F(s+ih_j(\xi)) \tag \(*\) \] with \(F_j\in\mathcal S\) and \(h_j(\xi)\) continuous. In the first theorem the authors prove the rigidity conjecture, assuming the Selberg orthonormality conjecture and the countability conjecture. This theorem has been independently obtained by \textit{U. Vorhauer} and \textit{E. Wirsing} in [On Sarnak's rigidity conjecture, J. Reine Angew. Math. 531, 35-47 (2001; Zbl 0960.11044)]. Further, they define an analytic family \({\mathcal F} =\{F(s;\xi)\}_{\xi\in\mathcal U}\), where \({\mathcal U}\supset{\mathbb R}\) is a domain in \(\mathbb C\), the function \(F(s;\xi)\) belongs to \(\mathcal S\) for \(\xi\in \mathbb R\) and \({\mathcal F}\) satisfies some natural holomorphicity conditions. For the analytic family \({\mathcal F}\) it is expected that the functions \(h_j(\xi)\) in \((*)\) can be chosen to be holomorphic on a suitable domain \({\mathcal V}\supset\mathbb R\). In this case, \({\mathcal F}\) is called an analytic shifted family. The second theorem states that, assuming the same conditions as in the first theorem, every analytic family is an analytic shifted family. Also the authors show that both theorems hold unconditionally in the class of automorphic \(L\)-functions.
scientific article; zbMATH DE number 1441964