THE VALUE DISTRIBUTION OF RANDOM DIRICHLET SERIES ON THE RIGHT HALF PLANE (II)

Abstract

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

CLC Number:

30B50

TIAN Fan-Ji, LIN Yao-Feng. THE VALUE DISTRIBUTION OF RANDOM DIRICHLET SERIES ON THE RIGHT HALF PLANE (II)[J].Acta mathematica scientia,Series B, 2003, 23(3): 426-.

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THE VALUE DISTRIBUTION OF RANDOM DIRICHLET SERIES ON THE RIGHT HALF PLANE (II)

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