Czechoslovak Mathematical Journal, Vol. 62, No. 4, pp. 919-936, 2012

Julia lines of general random Dirichlet series

Qiyu Jin, Guantie Deng, Daochun Sun

Qiyu Jin: Université de Bretagne-Sud, Campus de Tohaninic, BP 573, 56017 Vannes, France; Université Européne de Bretagne, France, e-mail: qiyu.jin@univ-ubs.fr; Guangtie Deng: Key Laboratory of Mathematics and Complex Systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People's Republic of China, e-mail: denggt@bnu.edu.cn; Daochun Sun: School of Mathematical Sciences, South China Normal University, Guangzhou 510631, People's Republic of China, e-mail: sundch@scnu.edu.cn

Abstract: In this paper, we consider a random entire function $f(s,\omega)$ defined by a random Dirichlet series $\sum\nolimits_{n=1}^{\infty}X_n(\omega) \ee^{-\lambda_n s}$ where $X_n$ are independent and complex valued variables, $0\leq\lambda_n \nearrow+\infty$. We prove that under natural conditions, for some random entire functions of order $(R)$ zero $f(s,\omega)$ almost surely every horizontal line is a Julia line without an exceptional value. The result improve a theorem of J. R. Yu: Julia lines of random Dirichlet series. Bull. Sci. Math. 128 (2004), 341-353, by relaxing condition on the distribution of $X_n$ for such function $f(s,\omega)$ of order $(R)$ zero, almost surely.

Keywords: random Dirichlet series, order $(R)$, Julia lines, entire function

Classification (MSC 2010): 30D35


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