On the binomial asymptotics of an entire Dirichlet series (Q2754805)

The author considers Dirichlet series NEWLINE\[NEWLINE F(s)=\sum_{n=0}^\infty a_n\exp (s\lambda_n),\quad s=\sigma +it, NEWLINE\]NEWLINE where \(\lambda_n\to \infty\). Let \(M(\sigma ,F)=\sup_{t\in \mathbb R}|F(\sigma +it)|\), \(\mu (\sigma ,F)=\max_{n\geq 0}|a_n|\exp (\sigma \lambda_n)\). Conditions are found under which the inequalities NEWLINE\[NEWLINE \log \mu (\sigma ,F)\leq \Phi_1 (\sigma)+(1+o(1))\tau \Phi_2 (\sigma) NEWLINE\]NEWLINE and NEWLINE\[NEWLINE \log M(\sigma ,F)\leq \Phi_1 (\sigma)+(1+o(1))\tau \Phi_2 (\sigma) NEWLINE\]NEWLINE with some functions \(\Phi_1,\Phi_2\) and \(\tau \in \mathbb R\) are equivalent.