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| ON THE GROWTH OF SOME RANDOM HYPERDIRICHLET SERIES |
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Citation： |
Liu Quan-sheng.ON THE GROWTH OF SOME RANDOM HYPERDIRICHLET SERIES[J].Chinese Annals of Mathematics B,1989,10(2):214~220 |
| Page view： 1029
Net amount： 790 |
Authors： |
Liu Quan-sheng; |
Foundation： |
Partially supported by the National Science Foundation of P. B. 0. |
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| Abstract： |
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. |
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