Behavior of logarithm of modulus of the sum of Dirichlet series on curves (Q1354704)

Let \(F(s)= \sum^\infty_{n=1} a_ne^{\lambda_ns}\) be an entire Dirichlet series \[ \sum^\infty_{r=1} {1\over\lambda^2_n} <\infty,\;Q(z)= \prod^\infty_{n=1} \left(1-{z^2 \over \lambda^2_n} \right),\;q_n= -\log\bigl|Q'(\lambda_n) \bigr|. \] Denote \(c(t)= \max_{q_n<1} q_n\). The main result of this paper is Theorem: Suppose that \(\int^\infty_1 {c(t) \over t^2} dt<\infty\). If \(\sum^\infty_{n=1} {1\over\lambda_n} <\infty\) then \(\varlimsup_{{s\in \gamma\atop s\to\infty}} {\log|F(s) |\over \log M(\text{Re} s)}=1\) \(M(\sigma)= \sup_{|t|<\infty} |F(\sigma+it) |\) where \(\gamma\) is a curve going to infinity in such a way that for \(s\in\gamma\), \(s\to\infty\) we have \(\text{Re} s\to+\infty\). The conditions in the theorem are weaker than the corresponding conditions of the criterion for the interpolation sequences. Since the interpolation method cannot be applied to a more general situation, the author uses the method of the theory of exponential series.