Limit theorems in the space of analytic functions for the Hurwitz zeta-function with an algebraic irrational parameter (Q539215)

The Hurwitz zeta-function \(\zeta(s,\alpha)\), \(s=\sigma+it\), with a fixed parameter \(\alpha\), \(0<\alpha \leq 1\), for \(\sigma>1\), is given by the series \[ \zeta(s,\alpha)=\sum_{m=0}^{\infty}\frac{1}{(m+\alpha)^s}, \] and by analytic continuation elsewhere except for a simple pole at \(s=1\). The authors prove the limit theorem in the sense of the weakly convergent probability measures for the function \(\zeta(s,\alpha)\) with algebraic irrational parameter \(\alpha\) in the space of analytic functions, i.e., it is shown that, for \(T>0\), the probability measure \(\frac{1}{T}\text{meas}\big\{\tau \in [0,T]: \zeta(s+i\tau,\alpha)\in A\big\}\), \(A \in {\mathcal B}(H(D_j))\), \(j=1,2\), converges weakly to the distribution of the random element \(\zeta_j(s,\alpha,\omega)\) as \(T \to \infty\). Here \(D_1=\{s \in \mathbb C: \tfrac{1}{2}<\sigma<1\}\) and \(D_2=\{s \in \mathbb C: \sigma>1\}\).