摘要
The paper considers the random L-Dirichlet seriesf（s,ω）=sum from n=1 to ∞ P<sub>n</sub>（s,ω）exp（-λ<sub>n</sub>s）and the random B-Dirichlet seriesψτ<sub>0</sub>（s,ω）=sum from n=1 to ∞ P<sub>n</sub>（σ+iτ<sub>0</sub>,ω）exp（-λ<sub>n</sub>s）,where {λ<sub>n</sub>} is a sequence of positive numbers tending strictly monotonically to infinity, τ<sub>0</sub>∈R is a fixed real number, andP<sub>n</sub>（s,ω）=sum from j=1 to m<sub>n</sub> ε<sub>nj</sub>a<sub>nj</sub>s<sup>j</sup>a random complex polynomial of order m<sub>n</sub>, with {ε<sub>nj</sub>} denoting a Rademacher sequence and {a<sub>nj</sub>} a sequence of complex constants. It is shown here that under certain very general conditions, almost all the random entire functions f（s,ω） and ψ<sub>τ<sub>0</sub></sub>（s,ω） have, in every horizontal strip, the same order, given byρ=lim sup(（λ<sub>n</sub>logλ<sub>n</sub>）/(log A<sub>n</sub><sup>-1</sup>))whereA<sub>n</sub>=max |a<sub>nj</sub>|.Similar results are given if the Rademacher sequence {ε<sub>nj</sub>} is replaced by a steinhaus seqence or a complex normal sequence.
The paper considers the random L-Dirichlet series f(s,ω)=sum from n=1 to ∞ P_n(s,ω)exp(-λ_ns) and the random B-Dirichlet series ψτ_0(s,ω)=sum from n=1 to ∞ P_n(σ+iτ_0,ω)exp(-λ_ns), where {λ_n} is a sequence of positive numbers tending strictly monotonically to infinity, τ_0∈R is a fixed real number, and P_n(s,ω)=sum from j=1 to m_n ε_(nj)a_(nj)s^j a random complex polynomial of order m_n, with {ε_(nj)} denoting a Rademacher sequence and {a_(nj)} a sequence of complex constants. It is shown here that under certain very general conditions, almost all the random entire functions f(s,ω) and ψ_(τ_0)(s,ω) have, in every horizontal strip, the same order, given by ρ=lim sup((λ_nlogλ_n)/(log A_n^(-1))) where A_n=max |a_(nj)|. Similar results are given if the Rademacher sequence {ε_(nj)} is replaced by a steinhaus seqence or a complex normal sequence.
基金
Partially supported by the National Science Foundation of P. R. C.
