The value distribution of random Dirichlet series on the right half plane (II)

Kahane has studied the value distribution of the Gauss-Taylor series Sigma(n=0)(infinity) a(n)X(n)z(n), where {X-n} is a Gauss sequence and Sigma(n=1)(infinity)\a(n)\(2) = infinity. 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 Sigma(n=0)(infinity)X(n)e(-lambdanS) is studied where {X-n} is a sequence of some non-degenerate independent random variable satisfying conditions: SigmaX(n) = 0; Sigma(n=0)(infinity) E\X-n\(2) = +infinity;For Alln is an element of N, X-n or ReXn or ImX(n) of bounded density. There exists alpha > 0 such that For Alln : alpha(2)E\X-n\(2) less than or equal to E-2\X-n\ < +infinity (the classic Gauss and Steinhaus randoii variables axe special cases of such random variables). The important results axe obtained that every point on the line Res = 0 is a Picard point of the series without finite exceptional value a.s.