Regularly increasing entire Dirichlet series. (Q1889512)

Let \(F(s)=\sum_{n=0}^\infty a_ne^{s\lambda_n}\), \(s=\sigma+it\in\mathbb C\), be an entire Dirichlet series with \(\lambda_0=0\leq\lambda_n\nearrow+\infty\). Put \(M(\sigma):=\max\{| F(\sigma+it)| : t\in\mathbb R\}\), \(\mu(\sigma):=\max\{| a_n| e^{\sigma\lambda_n}: n\geq0\}\). The authors prove that the function \(\log\mu\) is regularly varying of order \(\rho\in[1,+\infty)\) (i.e. \(\log\mu(\sigma,F)=x^\rho\alpha(\sigma)\), where \(\lim_{\sigma\to+\infty}\frac{\alpha(c\sigma)}{\alpha(\sigma)}=1\) for arbitrary \(c>0\)) iff there exists an increasing sequence \((n_k)_{k=0}^\infty\subset\mathbb Z_+\) such that: (i) \(\varkappa_k:=\frac{\log| a_{n_k}| -\log| a_{n_{k+1}}| }{\lambda_{n_{k+1}}- \lambda_{n_k}}\nearrow+\infty\), (ii) \(| a_n| e^{\varkappa_k\lambda_n}\leq| a_{n_k}| e^{\varkappa_k\lambda_{n_k}}\), \(n_k<n<n_{k+1}\), (iii) \(\frac{\varkappa_k\lambda_{n_{k+1}}}{\varkappa_k\lambda_{n_{k+1}}+ \log| a_{n_{k+1}}| }\to\rho\), (iv) \(\frac{\varkappa_k\lambda_{n_k}}{\varkappa_k\lambda_{n_k}+\log| a_{n_k}| } \to\rho\). The functions \(\log\mu\) and \(\log M\) need not be both regularly varying of the same order.