Rearrangement of coefficients for a Dirichlet series (Q2714883)

Let \(\{n'\}\) be a rearrangement of \(\{n\}^{+ \infty}_{n=0} \subset \mathbb{N}\), i.e., \(\{n'\}\) is formed from \(\{n\}\) by a bijection. Consider the Dirichlet series \(f(z)= \sum^{+\infty}_{n=0} a_ne^{\lambda_nz}\) and \(f_1(z)= \sum^{+\infty}_{n=0} a_{n'}e^{\lambda_nz}\) \((0\leq\lambda_n \uparrow+ \infty)\).NEWLINENEWLINENEWLINE1) If \(\varlimsup(\ln n)/ \lambda_n <+\infty\) and \(\lim (\ln|a_n |)/ \lambda_n= -\infty\) \((n\to +\infty)\), then \(\rho=\rho' \Leftrightarrow \lambda_{n'} \ln\lambda_{n'}= \lambda_n\ln \lambda_n+ o( \lambda_n \ln\lambda_n)\) \((n\to +\infty)\), where \(\rho=\varlimsup (\ln^+ \ln^+M(x,f))/ \ln x\) \((x\to+ \infty)\), \(M(x,f)=\max \{|f(x+iy) |:y \in\mathbb{R}\}\) and \(\rho'\) is defined analogously by \(f_1(z)\).NEWLINENEWLINENEWLINE2) If \(\varlimsup(\ln\ln n)/ \ln \lambda_n <\rho_1/(1+ \rho_1)\) and \(\varlimsup (\ln|a_n|)/ \lambda_n= 0\) \((n\to+ \infty)\), then \(\rho_1= \rho_1'\Leftrightarrow \ln\lambda_{n'}= \ln \lambda_n +o(\ln \lambda_n)\) \((n\to +\infty)\), where \(\rho_1= \varlimsup (\ln^+ \ln^+ M(x,f))/ \ln(-1/x)\) \((x\to-0)\) and \(\rho_1'\) is analogously defined by \(f_1(z)\).