Maximal term of the modified Dirichlet series (Q5961075)

Let \(0<\lambda_n\nearrow+\infty\) be such that \(\limsup_{n\to+\infty}\frac{\log n}{\log\lambda_n}<+\infty\). Let \(F(s)=\sum_{n=1}^\infty a_ne^{\lambda_ns}\), Re \(s<0\). For a sequence \((b_n)_{n=1}^\infty\), let \(F^\ast(s):=\sum_{n=1}^\infty a_nb_ne^{\lambda_ns}\). Put \(\mu(\sigma):=\max_{n\geq 1}\{|a_n|e^{\lambda_n\sigma}\}\), \(\mu^\ast(\sigma):=\max_{n\geq 1}\{|a_nb_n|e^{\lambda_n\sigma}\}\). The author characterizes those sequences \((b_n)_{n=1}^\infty\) for which \(\log\mu(\sigma)=(1+o(1))\log\mu^\ast(\sigma)\) as \(\sigma\longrightarrow 0-\), \(\sigma\in A\subset[-1,0)\), where the set \([-1,0)\setminus A\) is exceptional.