Complete equivalence of the logarithms of the maximum modulus and the maximal term of an entire Dirichlet series
Let \(\Lambda =(\lambda_ n)^{\infty}_{n=0}\), \(0\leq \lambda_ n\nearrow +\infty\) and let S(\(\Lambda\)) denote the set of all entire functions F of the form \(F(z)=\sum^{\infty}_{n=0}a_ n \exp (\lambda_ nz)\). Put \[ M(\sigma,F):=\sup \{| F(\sigma +it)|:\;t\in {\mathbb{R}}\},\quad \mu (\sigma,F):=\max \{| a_ n| \exp (\lambda_ n\sigma):\;n\geq 0\}. \] The author studies the problem of the asymptotic behaviour of the functions log M(\(\sigma\),F) and log \(\mu\) (\(\sigma\),F) when \(\sigma \to +\infty\). He proves that for any \(\Lambda\) there exists a function \(F\in S(\Lambda)\) for which the relation \[ (*)\quad \log M(\sigma,F)\sim \log \mu (\sigma,f),\quad \sigma \to +\infty, \] is not fulfilled. Moreover, he characterizes some classes of sequences \(\Lambda\) and functions \(F\in S(\Lambda)\) for which condition (*) is satisfied.
scientific article; zbMATH DE number 4165234