Growth of horizontal rays of analytic functions represented by Dirichlet series (Q1813646)

Given a Dirichlet series \(F(z)= \sum^a_n \exp (\lambda_nz)\) absolutely convergent in \(\text{Re} z<0\), we know that the Ritt order of \(F(z)\) in \(\text{Re} z<0\) is the same as that on every horizontal half-line in \(\text{Re} z<0\) if \(\sum(1/ \lambda_n) <+\infty\) and \(\liminf_{n\to +\infty} (\lambda_{n+1} -\lambda_n)>0\) [see the reviewer, C. R. Acad. Sci., Paris, Sér. I 296, 187-190 (1983; Zbl 0531.30006); also in Publ. Math. Orsay 83-02, 106-116 (1983; Zbl 0511.30004)]. The authors extend this result to the case of the generalized Ritt order \(=\lim_{\sigma\to-0} \alpha(\ln M(\sigma,F))/ \beta(1/ |\sigma |)\), where \(M(\sigma,F) =\sup_t \{| F(\sigma+it) |\}\) and \(\alpha\) and \(\beta\) are some continuous and increasing functions. They utilize the finite index of condensation of \(\{\lambda_n\}\), obtain some sufficient conditions with a necessary character and construct Dirichlet series absolutely convergent in \(\text{Re} z>0\) and bounded on the negative axis.