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 t...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.展开更多
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 Sig...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.展开更多
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 {...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.展开更多
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
This paper deals with the value distribution of random Dirichlet series whose coefficients are a martingale difference sequence, and which is of neutral growth.
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 > ...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.展开更多
In this article, the uniqueness theorem of Dirichlet series is proved. Then the random Dirichlet series in the right half plane is studied, and the result that the random Dirichlet series of finite order has almost su...In this article, the uniqueness theorem of Dirichlet series is proved. Then the random Dirichlet series in the right half plane is studied, and the result that the random Dirichlet series of finite order has almost surely（a.s.） no deficient functions is proved.展开更多
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?lde...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.展开更多
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 ineq...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..展开更多
文摘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.
文摘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.
基金Supported by the National Natural Science Foundation of China（10471048）
文摘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.
基金the Doctoral Programme Fundation and by theNational Natural Science Fundation of China
文摘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
基金the National Natural Science Foundation of China and the Doctoral Foundation of China.
文摘This paper deals with the value distribution of random Dirichlet series whose coefficients are a martingale difference sequence, and which is of neutral growth.
基金Project supported by the National Natural Science Foundationof China
文摘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.
文摘In this article, the uniqueness theorem of Dirichlet series is proved. Then the random Dirichlet series in the right half plane is studied, and the result that the random Dirichlet series of finite order has almost surely（a.s.） no deficient functions is proved.
文摘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.
文摘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..
