On the distribution of values of random Dirichlet series. II (Q1812863)

Let \(\{Z_ n(\omega)\}\) be a complex or real random variable sequence of independent, symmetric and equally distributed and finite variance, for which \(\exists k\in\mathbb{N}\) such that \[ \int_{| Z|<1}| Z|^{-{1\over k}}\mu(dZ), \] where \(\mu\) is the common measure defined by \(Z_ n(\omega)\) (it includes the classical Rademacher, Steinhaus and Gauss sequences). Suppose that the random Dirichlet series \[ f_ \omega(s)=\sum_{n=1}^ \infty b_ nZ_ n(\omega)e^{- \lambda_ n s} \] satisfies \[ \overline {\lim}_{n\to\infty}{\log \log n \over \log \lambda_ n}={\rho \over \rho+1} (0<\rho<\infty),\qquad \overline {\lim}_{n\to\infty} {\log^ + \log^ +| b_ n| \over \log \lambda_ n}={\rho \over \rho+1}, \] then i) the order of \(f_ \omega(s)\) are almost surely \(\rho\). ii) Every point of the imaginary axis is almost surely a Borel point of order \(\rho+1\) of \(f_ \omega(s)\) and with no finite exceptional value. In the part (I) of this paper [\textit{Yu Jiarong} and \textit{Sun Daochun}, Lect. Complex analysis, Singapore, World Scientific Publishing, 1988, 67-95 (1988)], the case of infinite order was discussed.