Asymptotics of the logarithm of the maximal term of the modified Dirichlet series (Q1397547)

Let a Dirichlet series \(F(s)=\sum^\infty_{n=1} a_ne^{\lambda_n s}\) be an entire function of finite Ritt's order. The authors have considered the series \(F^*(s)=\sum^\infty_{n=1} a_nb_ne^{\lambda_ns}\) where the sequence of complex numbers \(\{b_n\}\), \(b_n\neq 0\) for \(n\geq N\), satisfies the condition \(\limsup_{n\to\infty} {\ln|b_n|\over\lambda_n} <\infty\). The main result of the paper is Theorem 2 which establishes the asymptotic equality between the maximal terms of the series \(F(s)\) and \(F^*(s)\) outside of any exceptional set.