Generalized and modified orders of growth for Dirichlet series absolutely convergent in a half-plane

Abstract

Let $\lambda=(\lambda_n)_{n\in\mathbb{N}_0}$ be a non-negative sequence increasing to $+\infty$, $\tau(\lambda)=\varlimsup_{n\to\infty}(\ln n/\lambda_n)$, and $\mathcal{D}_0(\lambda)$ be the class of all Dirichlet series of the form $F(s)=\sum_{n=0}^\infty a_n(F)e^{s\lambda_n}$ absolutely convergent in the half-plane
$\operatorname{Re}s<0$ with $a_n(F)\not=0$ for at least one integer $n\ge0$. Also, let $\alpha$ be a continuous function on $[x_0,+\infty)$ increasing to $+\infty$, $\beta$ be a continuous function on $[a,0)$ such that $\beta(\sigma)\to+\infty$ as $\sigma\uparrow0$, and $\gamma$ be a continuous positive function on $[b,0)$. In the article, we investigate the growth of a Dirichlet series $F\in\mathcal{D}_0(\lambda)$ depending on the behavior of the sequence $(|a_n(F)|)$ in terms of its $\alpha,\beta,\gamma$-orders determined by the equalities
$$R^*_{\alpha,\beta,\gamma}(F)=\varlimsup_{\sigma\uparrow0} \frac{\alpha(\max\{x_0,\gamma(\sigma)\ln\mu(\sigma)\})}{\beta(\sigma)},$$
$$R_{\alpha,\beta,\gamma}(F)=\varlimsup_{\sigma\uparrow0} \frac{\alpha(\max\{x_0,\gamma(\sigma)\ln M(\sigma)\})}{\beta(\sigma)},$$
where $\mu(\sigma)=\max\{|a_n(F)|e^{\sigma\lambda_n}\colon n\ge0\}$ and $M(\sigma)=\sup\{|F(s )|\colon \operatorname{Re}s=\sigma\}$ are the maximal term and the supremum modulus of the series $F$, respectively. In particular, if for every fixed $t>0$ we have $\alpha(tx)\sim \alpha(x)$ as $x\to+\infty$, $\beta(t\sigma)\sim t^{-\rho}\beta(\sigma)$ as $\sigma\uparrow0$ for some fixed $\rho>0$, $0<\varliminf_{\sigma\uparrow0}\gamma(t\sigma)/\gamma(\sigma) \le \varlimsup_{\sigma\uparrow0}\gamma(t\sigma)/\gamma(\sigma)<+\infty$,
$\Phi(\sigma)=\alpha^{-1}(\beta(\sigma))/\gamma(\sigma)$ for all $\sigma\in[\sigma_0,0)$,
$\widetilde{\Phi}(x)=\max\{x\sigma-\Phi(\sigma)\colon \sigma\in[\sigma_0,0)\}$ for all $x\in\mathbb{R}$, and $\Delta_\Phi(\lambda)=\varlimsup_{n\to\infty}( -\ln n/\widetilde{\Phi}(\lambda_n))$, then:

(a) for each Dirichlet series $F\in\mathcal{D}_0(\lambda)$ we have
$$R^*_{\alpha,\beta,\gamma}(F)=\varlimsup_{n\to +\infty}\left(\frac{\ln^+|a_n(F)|}{-\widetilde{\Phi }(\lambda_n)}\right)^\rho;$$

(b) if $\tau(\lambda)>0$, then for each $p_0\in[0,+\infty]$ and any positive function $\Psi$ on $[c,0)$ there exists a Dirichlet series $F\in\mathcal{D}_0(\lambda)$ such that $R^*_{\alpha,\beta,\gamma}(F)=p_0$ and $M(\sigma,F)\ge \Psi(\sigma)$ for all $\sigma\in[\sigma_0,0)$;

 (c) if $\tau(\lambda)=0$, then $(R_{\alpha,\beta,\gamma} (F))^{1/\rho}\le (R^*_{\alpha,\beta,\gamma}(F))^{1/\rho}+\Delta_\Phi(\lambda)$ for every Dirichlet series\linebreak $F\in\mathcal{D}_0(\lambda)$;

 (d) if $\tau(\lambda)=0$, then for each $p_0\in[0,+\infty]$ there exists a Dirichlet series $F\in\mathcal{D}_0(\lambda)$ such that $R^*_{\alpha,\beta,\gamma}(F)=p_0$ and $(R_{\alpha,\beta,\gamma}(F))^{1/\rho}=(R ^*_{\alpha,\beta,\gamma}(F))^{1/\rho}+\Delta_\Phi(\lambda)$.