The Lower Orders of Dirichlet and Random Dirichlet Series

The formulas of the lower orders of Dirichlet series are given by means of the exponents and the convex regularized sequences of the logarithms and the convex regularized sequences of the logarithms of the moduli of the coefficients. Corresponding results are obtained for some random Dirichlet series.

THE GROWTH OF RANDOM DIRICHLET SERIES (I)

Under the conditions(without independence): (i) There Exists alpha > 0, such that sup E\Z(n)\(alpha) < +infinity, (ii) There Exists beta > 0, such that sup E\Z(n)\(-beta) < +infinity, the random series Sigma a(n) Z(n)e(-lambda n) and series' Sigma a(n)e(-lambda ns) a.s. have the same abscissa of convergence, the (R) order, lower order and type.

The Growth of Dirichlet Series and Random Dirichlet Series of Zero Order and Finite Order

In this paper,we study the relations between the coefficients and the growth of zero order and finite order Dirichlet series and random Dirichlet series in the whole plane. And when the random variable sequence {Xn {ω}} satisfies the certain condition, in the whole plane, the growth of the random entire function which is determined by the zero order and finite order random Dirichlet series is almost surely same with corresponding growth of random Dirichlet series on any horizontal straight line.

BOREL LINES OF RANDOM DIRICHLET SERIES

For some random Dirichlet series of order (R) infinite almost surely every horizontal line is a Borel line of order (R) infinite and without exceptional values

Random Dirichlet Series with Coefficients Satisfying only a Moment Condition

This note implies only a moment condition upon the coefficients of random Dirichlet series to study the convergence and growth of the series. The condition needs the coefficients to satisfy the so-called inverse H?lder inequality, which need not be independent. The note uses a method whose feature is to compare the convergence of two series, and obtains two theorems, one dealing with the convergence of the random Dirichlet series, another the growth of the random analytic function represented by the series. These results can be used to improve essentially some known conclusions.

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

Kahane has studied the value distribution of the Gauss-Taylor serieswhere {Xn} is a complex Gauss sequence and In this paper, by trans-forming the right half plane into the unit disc and setting up some important inequal-ities, the value distribution of the Dirichlet series is studied where {Xn}is a sequence of some non-degenerate independent random variable satisfying conditions:EXn = 0; Xn or ReXn or ImAn of bounded density. Thereexists α> 0 such that Vn : (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 Res = 0 is a Picard point of the series without finite exceptional value a.s..

THE VALUE DISTRIBUTION OF RANDOM ANALYTIC DIRICHLET SERIES OF NEUTRAL GROWTH (Ⅱ)

This paper studies the value distribution of random analytic Dirichlet series f(s) = Zn()e-sn, where {Zn} is a sequence of independent random variables, n = 1 with moments zero, such that infE{Zn}/E1/2{Zn2≥ α > 0. Suppose [h*(σ)]2 = n converges for any α > 0, and diverges for = 0. It is shown that if = ρ E (0, ), then with probability one, where β is a constant depending only upon the constant α.

The Growth of Random Dirichlet Series

For a known random Dirichlet series of infinite order on the whole plane, the authors construct a Dirirchlet series such that the growth of both series referring to the type function is the same. Thus one can study the growth of the former by studying the coefficient and exponent of the latter.