Joint discrete limit theorems on the complex plane for general Dirichlet series (Q2571498)

Extending a former joint paper with [\textit{A. Laurinčikas} and {R. Macaitienė}, ``Discrete limit theorems for general Dirichlet series'', Chebyshevskii Sb. 4, No. 3, 156--170 (2003; Zbl 1105.11030)], the authoress is going to prove discrete \textit{joint}\ limit theorems for \(n\) general Dirichlet series \[ f_j(s) = \sum_1^\infty a_{m,j} e^{-\lambda_m s}, \text{ for } \sigma > \sigma_{a,j}, \; j=1,2,\dots,n \] (for notations and assumptions see the preceding review of the paper of A. Laurinčikas; in particular, the \(f_j(s)\) are supposed to be meromorphically continuable to the half--plane \(\sigma > \sigma_{1,j}\), with all poles contained in some compact set). If \(h>0\) is fixed, then the probability measures \(Q_N(A)\) \(\bigl(\)defined on Borel sets \(A\subset \mathbb C^n \bigr)\) \[ \begin{gathered} Q_N(A)\! = \!\!\mu_N\Bigl( \bigl( f_1(\sigma_1+imh), \dots, f_n(\sigma_n +imh) \bigr) \in A \Bigr) \\ \Bigl[\! = \!\!\frac1{N+1}\cdot \# \left\{ 0\leq m \leq N;\,\Bigl( \bigl( f_1(\sigma_1+imh), \dots, f_n(\sigma_n +imh) \bigr) \in A \Bigr) \right\} \Bigr] \end{gathered} \] converge weakly, as \(N\to\infty\), to some probability measure \(Q\) on the Borel sets of \(\mathbb C^n\) (if \(\sigma_j > \sigma_{1,j}\)). If the \(\{\lambda_m\}\) is an increasing sequence of \textit{algebraic, \(\mathbb Q\)-linearly independent}\ positive numbers, satisfying \(\lambda_m \geq c\cdot (\log m)^\delta\), and if \(\exp\left( \frac{2\pi}h\right) \in \mathbb Q\), then the limit measure \(Q\) is equal to the distribution \(Q_F\) of the \(\mathbb C^n\)-valued random element \(F\), \[ F(\sigma_1,\dots,\sigma_n;\,\omega) = \left( \sum_{m=1}^\infty a_{m,1} \omega(m) e^{-\lambda_m \sigma_1}, \dots, \sum_{m=1}^\infty a_{m,n} \omega(m) e^{-\lambda_m \sigma_n} \right). \]