ON THE GROWTH OF SOME RANDOM HYPERDIRICHLET SERIES

The paper considers the random L-Dirichlet seriesf(s,ω)=sum from n=1 to ∞ P_n(s,ω)exp(-λ_ns)and the random B-Dirichlet seriesψτ_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, andP_n(s,ω)=sum from j=1 to m_n ε_(nj)a_(nj)s~ja 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 byρ=lim sup((λ_nlogλ_n)/(log A_n~(-1)))whereA_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.

ON THE GROWTH OF SOME RANDOM HYPERDIRICHLET SERIES

The paper considers the random L-Dirichlet series $$\f(s,w) = \sum\limits_{n = 1}^\infty {{P_n}(s,w)\exp ( - {\lambda _n}s)} \]$$ and the random B-Dirichlet series $$\{\varphi _{{\tau _0}}}(s,w) = \sum\limits_{n = 1}^\infty {{P_n}(\sigma + i{\tau _0},w)\exp ( - {\lambda _n}s)} \]$$ where ${\{{\lambda _n}}\]}$ is a sequence of positive numbers tending strictly monotonically to infinity,$\{\tau _0} \in R\]$ is a fixed real number, and $$\{P_n}(s,w) = \sum\limits_{j = 0}^{{m_n}} {{\varepsilon _{nj}}{a_{nj}}{s^j}} \]$$ a random complex polynomial of order $\{m_n}\]$, with ${\{\varepsilon _{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,w)\]$ and $\{\varphi _{{\tau _0}}}(s,w)\]$ have, in every horizontal strip, tke same order, given by $$\\rho = \lim \sup \frac{{{\lambda _n}\log {\lambda _n}}}{{\log A_n^{ - 1}}}\]$$ where $$\{A_n} = \mathop {\max }\limits_{0 \le j \le {m_n}} \left| {{a_{nj}}} \right|\]$$ Similar results are given if the Rademacber sequence $\\{ {\varepsilon _{nj}}\} \]$ is replaced by a steinhaus seqence or a complex normal sequence.