Discrete limit theorems for general Dirichlet series. II (Q1881758)

A Dirichlet series \(f(s)=\sum_{m=1}^\infty a_m e^{-\lambda_m s}\) is considered with real \(\lambda_m>c(\log m)^\delta\), \(f(\sigma+it)=O(| t| ^\alpha)\), \(\alpha>0\) as \(| t| \to\infty\), \(\int_{-T}^T| f(\sigma+it)| ^2\,dt=O(T) \to\infty\). Denote \[ \mu(A)={1\over N+1}\text{card}\{f(\sigma+imh)\in A;\;m=0,1,\dots, N\}. \] It is shown that if \(\{\lambda_m\}\) are linearly independent over the field of rational numbers, then \(\mu(A)\) converges weakly to \(\{\tilde f(s)\in A\}\), where \(\tilde f(s)=\sum_{m=1}^\infty a_m\xi_m e^{-\lambda_m s}\), \(\xi_m\) being independent uniformly distributed on the complex unit circle. [For part III see Cent. Eur. J. Math. 2, No.~3, 339--361 (2004; Zbl 1109.11042)]