The Nature Of A Kind Of Random Series On The Plane

This paper first gives the definition of pairwise (?)-N sequences and generalizes Borel-Cantelli lemma to pairwise (?)-N sequences.Then,by using the generalized Borel-Cantelli lemma,we mainly discuss the problems about the growth and the rearrangements of the random Taylor series whose coefficients are pairwise (?)-N sequences. Besides,using the generalized Borel-Cantelli lemma,we also discuss the problems about the convergence and growth of double random Dirichlet series whose coefficients and indexs are pairwise (?)-N sequences.This paper consists of four parts:The first part is introduction.This part mainly introduces the background of Dirich-let series,research status at home and abroad of Dirichlet series and the main content and research results of this paper.The second part is the pairwise (?)-N sequence.This part firstly presents the definition of (?)-N sequence.then using the pairwise (?)-N sequence generalize Borel-Cantelli lemma.The third part is the random Taylor series.By using the generalized Borel Cantelli lemma,we mainly discuss the problems about the growth and the rearrange-ments of the random Taylor series whose coefficients are pairwise (?)-N sequences in this part,we get this kind of random Taylor series and some series have the same growth,as well as its growth and type remain constant after rearrangement of coeffi-cients under certain conditions.The fourth part is the double random Dirichlet series.Using the generalized Borel-Cantelli lemma,we also discuss the problems about the convergence and growth of double random Dirichlet series whose coefficients and indexs are pairwise (?)-N sequences,we get the simple formula of abscissa of convergence and prove that this kind of double random Dirichlet series and Dirichlet series have the same growth,as well as the (p, q)(R)-...