On Sarnak's rigidity conjecture (Q2703577)

In 1989 \textit{A. Selberg} [Proc. Amalfi Conf. Analytic Number Theory, Univ. Salerno, 367-385 (1992; Zbl 0787.11037); see also Collected Papers, Vol. II, pp. 47-63 (1991; Zbl 0729.11001)] defined a class \({\mathcal S}\) of functions, in an attempt to characterize those functions for which one expects a Riemann Hypothesis to be true. Selberg's conditions involved such things as a functional equation and an Euler product. NEWLINENEWLINENEWLINEA function \(F\) in \({\mathcal S}\) is called primitive, if it cannot be written as a nontrivial product of two elements of \({\mathcal S}\). If \(F(s)\) is an entire function in \({\mathcal S}\), then for any real \(\theta\) the shifted function \(F(s+ i\theta)\) is also in \({\mathcal S}\). Functions \(F(s,\xi)\in{\mathcal S}\) depending on a real parameter \(\xi\) are called a continuous (holomorphic) family, if \(F(s,\xi)\) is continuous (holomorphic) as a function of \(\xi\) for each \(s\neq 1\). We say a family is primitive if \(F(s,\xi)\) is primitive for each \(\xi\). NEWLINENEWLINENEWLINEA number of conjectures have been made related to \({\mathcal S}\). Selberg himself conjectured a certain form of orthogonality of different primitive members of the class. Secondly, it has been conjectured that there are only countably many shift classes of primitive functions in \({\mathcal S}\). Thirdly, Sarnak's Rigidity Conjecture asserts that any continuous family on an interval \(I\subset \mathbb{R}\) is obtained by continuously and independently shifting some primitive elements of \({\mathcal S}\) and then multiplying them. NEWLINENEWLINENEWLINEWe show that this third conjecture follows from the first two conjectures. The proof depends on an old lemma of Sierpiński concerning the topology of the real line. NEWLINENEWLINENEWLINEAn analogue for holomorphic families is also derived from the same two conjectures.