A joint limit theorem for the Riemann zeta-function in the space of analytic functions
Let \(H(D)\) denote the space of functions analytic in the critical strip \(D= \{ s\in \mathbb C: 1/2 < \Re s < 1 \}\) equipped with the topology of uniform convergence on compacta, and write the \(r\)-fold Cartesian product of \(H(D)\) by itself as \(H^{r}(D)\). By \(\mathcal B (S)\) denote the Borel sets of a space \(S\). Define the probability measure on \((H^{r}(D), \mathcal B (H^{r}(D)))\) as \[ P_{T}(A) = {1\over T}\operatorname {meas} \{\tau \in [0,T] ; (\zeta(s+ ik_{1}\tau),\ldots, \zeta(s+ ik_{r}\tau)) \in A \} , \] where the Lebesgue measure is meant and \(k_{j}\)'s are positive integers. In this paper it is proved that as \(T\to\infty\), \(P_{T}\) weakly converges to a probability measure \(P_{\zeta}\). The study is pursued in another paper [Lith. Math. J. 42, No. 4, 419-434 (2002; Zbl 1023.11042)] of the author where the explicit form of the limit measure \(P_{\zeta}\) is found.
scientific article; zbMATH DE number 1912624