A joint limit theorem for general Dirichlet series (Q5970577)

Let be given a collection of Dirichlet series \((s)=\sum_{m=1}^\infty a_{mj} e^{-\lambda_{mj}s}\) where \({mj}\) and \(\lambda_{mj}\) are real, \(\lambda_{mj}>C_j(\log m)^{\theta_j}\) for some \(\theta_j\). It is shown that if \(\lambda_{jm}\) are linearly independent over the field of rational numbers, then the measure \((A)={1\over T}\text{meas}\{(f_1(s_1+i\tau),\dots,f_n(s_n+i\tau))\in A)\}\) converges weakly to \(\{(\tilde f_i(s_i),i=1,\dots,n)\in A\}\) where \(\tilde f_i(s_i)=\sum_{m=1}^\infty a_{mj}\xi_{mj}e^{-\lambda_{mj}s}\), \(\xi_{jm}\) being independent uniformly distributed on the complex unit circle.