We study the growth of a Dirichlet series \( F(s)={\sum}_{n=1}^{\infty }{f}_n\exp \left\{s{\uplambda}_n\right\} \) with zero abscissa of absolute convergence with respect to the entire Dirichlet series \( G(s)={\sum}_{n=1}^{\infty }{g}_n\exp \left\{s{\uplambda}_n\right\} \) by using generalized quantities of an order \( {\upvarrho}_{\beta, \beta}^0{\left[F\right]}_G=\lim\ {\sup}_{\sigma \uparrow 0}\frac{\beta \left({M}_G^{-1}\left({M}_F\left(\sigma \right)\right)\right)}{\beta \left(1/\left|\sigma \right|\right)} \) and a lower order \( {\uplambda}_{\beta, \beta}^0{\left[F\right]}_G=\lim\ {\operatorname{inf}}_{\sigma \uparrow 0}\frac{\beta \left({M}_G^{-1}\left({M}_F\left(\sigma \right)\right)\right)}{\beta \left(1/\left|\sigma \right|\right)}, \) where \( {M}_F\left(\sigma \right)=\sup \left\{\left|F\left(\sigma + it\right)\right|:t\in \mathrm{\mathbb{R}}\right\},{M}_G^{-1}(x) \) is the function inverse to M_G(σ), and β is a positive function increasing to +∞.

我们研究狄利克雷级数的增长\( F(s)={\sum}_{n=1}^{\infty }{f}_n\exp \left\{s{\uplambda}_n\right\} \)相对于整个狄利克雷级数绝对收敛的零横坐标\( G(s)={\sum}_{n=1}^{\infty }{g}_n\exp \left\{s{\ uplambda}_n\right\} \)通过使用订单的广义数量\( {\upvarrho}_{\beta, \beta}^0{\left[F\right]}_G=\lim\ {\sup} _{\sigma \uparrow 0}\frac{\beta \left({M}_G^{-1}\left({M}_F\left(\sigma \right)\right)\right)}{\beta \left(1/\left|\sigma \right|\right)} \)和一个低阶\( {\uplambda}_{\beta, \beta}^0{\left[F\right]}_G=\lim\ {\operatorname{inf}}_{\sigma \uparrow 0}\frac{\beta \left({M}_G^{-1}\left({M}_F\left(\sigma \right)\right)\right)}{\beta \left(1/\left|\sigma \right| \right)}, \)其中\( {M}_F\left(\sigma \right)=\sup \left\{\left|F\left(\sigma + it\right)\right|:t\in \mathrm{\mathbb{R}}\right\},{M}_G^{-1}(x) \)是M _G ( σ ) 的逆函数，β是增加到 +∞ 的正函数。