In this paper, we consider a random entire function $f(s,\omega )$ defined by a random Dirichlet series $\sum \nolimits _{n=1}^{\infty }X_n(\omega ) {\rm e} ^{-\lambda _n s}$ 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.
In the paper we obtain that, under some condition, the Rademacher-Dirichlet series or the Steinhaus-Dirichlet series on the whole plane and on the horizontal zone almost surely have the same growth.
We define Knopp-Kojima maximum modulus and the Knopp-Kojima maximum term of Dirichlet series on the right half plane by the method of Knopp-Kojima, and discuss the relation between them. Then we discuss the relation between the Knopp-Kojima coefficients of Dirichlet series and its Knopp-Kojima order defined by Knopp-Kojima maximum modulus. Finally, using the above results, we obtain a relation between the coefficients of the Dirichlet series and its Ritt order. This improves one of Yu Jia-Rong's results, published in Acta Mathematica Sinica 21 (1978), 97–118. We also give two examples to show that the condition under which the main result holds can not be weakened.