The growth of Dirichlet series (Q5906754)

Consider the Dirichlet series \[ f(s)= \sum_{j>0} a_j\exp (-\lambda_js) \quad (0<\lambda_j \uparrow+ \infty,\;s\in\mathbb{C}), \tag{1} \] where \(\varlimsup(\ln n/ \lambda_n)= \varlimsup (\ln|a_n|/ \lambda_n) =0\) \((n\to +\infty)\). Then \(f(s)\) is analytic in \(\text{Re} s>0\). Let \(M(\sigma)= \sup \{|f(\sigma+it) |: \sigma>0,\;t\in\mathbb{R}\}\), \(m(\sigma)= \{|a_j|\exp(-\lambda_j \sigma): j\in\mathbb{N}_+\}\) and let \(U(r)= r^{\rho(r)}\) satisfy i) \(\rho(r) \downarrow 0\) and \(\rho'(r)\ln r\to 0\) \((r\to+0)\) and ii) \(U(r) \uparrow+ \infty\), \(U(kr) \leq(1+ o(1)) U(r)\) and \(U(r^k)=U^k(r)\) \((r\to \infty,\;k>1)\). The author proves that under the condition \(\varlimsup [\ln n/U(\lambda_n/ \ln n)]=0\) \((n\to\infty)\) 1) \(\varlimsup [\ln M(\sigma)/U(1/ \sigma)] =A\) \((\sigma\to +0)\Leftrightarrow \varlimsup\ln [|a_n|'/U (\lambda_n/ \ln|a_n |') =A\) \((n\to\infty)\), where \(|a_n|'= \max\{e,|a_n|\}\) and 2) \(\lim[\ln M (\sigma)/U(1/ \sigma)]=A\) \((\sigma\to 0) \Leftrightarrow\varlimsup [\ln|a_n|'/U (\lambda_n/\ln|a_n|') =A\) \((n\to\infty)\) and \(\exists \{\lambda_{n(p)}\} \subset \{\lambda_n\}\) such that \(\lim [\ln|a_{n(p)} |'/U(\lambda_{n(p)}/\ln|a_{n(p)} |')] =A\) \((p\to\infty)\) and \(\lim[U(h(\lambda_{n(p)}))/U (h(\lambda_{n(p+1)}))] =1\) \((p\to\infty)\), where \(u=h(v)\) is the inverse function of \(v=uU(u)\). Some analogous results for the limit and the upper limit of \([\ln\ln M(\sigma)/ \ln U(1/\sigma)]\) \((\sigma\to 0)\) are also obtained.