关键词: Random Dirichlet series, characteristic function, Picard point

Abstract:

Kahane has studied the value distribution of the Gauss-Taylor series
$\sum\limits^\infty_{n=0}a_nX_nz^n$,
 where $\{X_n\}$  is a complex Gauss sequence and
$\sum\limits^\infty_{n=1}|a_n|^2=\infty$.
In this paper, by  transforming the
 right half plane into the unit disc and
setting up some important inequalities,
 the value distribution of the Dirichlet series
$\sum\limits^\infty_{n=0}X_n{\rm e}^{-\lambda_nS}$
 is studied where $\{X_n\}$ is a sequence of some non-degenerate
independent random variable satisfying conditions:
$EX_n=0; \sum\limits^\infty_{n=0}E|X_n|^2=+\infty;
\forall n\in N, X_n$ or Re$X_n$ or Im$X_n$ of bounded density.
There exists $\alpha>0$ such that $\forall n:\alpha^2E|X_n|^2\leq E^2
|X_n|<+\infty$  (the classic Gauss and Steinhaus random variables
are special cases of such random variables).
The important results are obtained that every point on the line Re$s=0$ is
a Picard point of the series without finite exceptional value a.s..

Key words: Random Dirichlet series, characteristic function, Picard point