On certain relations between the maximum modulus and the maximal term of an entire Dirichlet series (Q1972530)

Let \(F(z)=\sum_{n=0}^\infty a_n\exp(z\lambda_n)\) be an entire function, \(0\leq\lambda_n\nearrow+\infty\). Put \(M(\sigma):=\sup\{|F(\sigma+it)|\:t\in\mathbb R\}\), \(\mu(\sigma):=\max\{|a_n|\exp(\sigma\lambda_n)\: n\geq 0\}\). Let \(\psi\) be an increasing function with \(x\leq\psi(x)\leq\exp(x)\), \(x\geq 0\). The author presents conditions under which \(\psi(\log M(\sigma))\sim\psi(\log\mu(\sigma))\) when \(\sigma\longrightarrow+\infty\) outside a set of finite measure.