On the regular growth of Dirichlet series absolutely convergent in a half-plane (Q765544)

Let \((\lambda_n)\) be a sequence of nonnegative numbers increasing to \(+\infty\) and \(f(s)=\sum_{n\geq 1}a_ne^{s\lambda_n}\) be a Dirichlet series with coefficients \(a_n\) belonging to \(\mathbb{C}\) and \(s=\sigma +it\). Assume that the abscissa of absolute convergence of \(f\) is equal to zero. The aim of the paper is to give conditions for \((\lambda_n)\) and \((a_n)\) under which \[ \log M(\sigma,f)=T_R(1+o(1))e^{\rho_R/| \sigma|} \] for \(\sigma\rightarrow 0\), \(\sigma<0\), where \(T_R\) and \(\rho_R\) are specific positive constants and \(M(\sigma,f)=\sup\{| f(\sigma +it)|:t\in\mathbb{R},\sigma<0\}\).