多重随机Dirichlet级数及其应用

Dirichlet series plays an important role in analytic number theory. Various forms of Dirichlet series (including random Dirichlet series) are related to many branches of mathematics. The main purpose of this program is to study the value distrbution of multiple Dirichlet series, random (simple) Dirichlet series and random multiple Dirichlet series, by using the methods in theory of complex function and the theory of probability. Then as an application of the results above,we will study the value distribution of random meromorphic functions.. Firstly,we will investigate the convergence of multiple Dirichlet series.The relation between the growth of multiple Dirichlet series and their coefficients will be obtained, and the existence of singular directions will be studied.. Secondly,by introducing a proper sequence of random variable, we will study some properties of random (simple) Dirichlet series and random multiple Dirichlet series, such as convergence,growth and the existence of singular directions..Thirdly, by the properties obtained in point 2, we continue to discuss the existence of singular direciotns of random meromorphic functions defined by random Dirichlet seires.

Dirichlet级数在解析数论中有重要的地位，其各种形式（包括精确的或随机的）的推广与众多数学分支有密切关系。因此相关研究具有意义。本项目将综合应用函数论和概率论的方法，研究多重Dirichlet级数，(简单)随机Dirichlet级数和多重随机Dirichlet的值分布。进一步，将上述结果应用到随机亚纯函数中去。主要研究内容包括：.1）研究多重Dirichlet级数的收敛性，得到级数的三种收敛横坐标精确表达式，在较弱的条件下讨论各种增长级与系数间的关系，并讨论级数各奇异方向的存在性；.2）引入适当的非独立非同分布的随机变量序列，研究（简单）随机Dirichlet级数和多重随机Dirichet级数的收敛性，增长性和各奇异方向的存在性；.3）利用2中（简单）随机Dirihclet级数的性质，研究由（简单）随机Dirichlet级数所决定的随机亚纯函数各奇异方向。

Dirichlet级数最初来源于解析数论,其本质上是一类全纯函数，其性质与其系数密切相关。而Laplace-Stieltjes变换和Laplace变换是Dirichelt级数的推广，它作为一种数学工具在数字信号处理中起重要作用。本项目已经完成研究内容：1.多重Dirichlet级数的增长性与系数的关系；2.Dirichlet级数的推广——Laplace-Stieltjes变换的广义级与型和系数的关系；3.代数体函数的值分布问题。