On a certain set of Lerch’s zeta-functions and their derivatives
On a certain set of Lerch’s zeta-functions and their derivatives
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关于一组 Lerch 的 zeta 函数及其导数
DOI：
10.1007/s10986-019-09433-0
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发表时间：
2019
影响因子：
0.4
通讯作者：
Hirofumi Nagoshi
中科院分区：
文献类型：
--
作者：
Hirofumi Nagoshi;Takashi Nakamura;Hirofumi Nagoshi
For 0< α≤ 1 and 0< λ≤ 1, letL(α, λ, s) denote the associated Lerch zeta-function, which is defined as $$ {\sum}_{n=0}^{\infty }{\mathrm{e}}^{2\pi \mathrm{i}n\uplambda}{\left(n+\alpha \right)}^{-s} $$ for ℜs >1. We investigate the joint value-distribution for all Lerch zeta-functions {L(αj, λ, s): 0< λ≤ 1, j= 1, … , J} and their derivatives whenα1,…, αJsatisfy a certain condition. This condition is satisfied ifα1,…, αJare algebraically independent over ℚ. More precisely, we establish a joint denseness result for values of those functions on vertical lines in the strip 1/2 < ℜs≤ 1. We also establish the functional independence of those functions in the sense of Voronin.
For 0< α≤ 1 and 0< λ≤ 1, letL(α, λ, s) denote the associated Lerch zeta-function, which is defined as $$ {\sum}_{n=0}^{\infty }{\mathrm{e}}^{2\pi \mathrm{i}n\uplambda}{\left(n+\alpha \right)}^{-s} $$ for ℜs >1. We investigate the joint value-distribution for all Lerch zeta-functions {L(αj, λ, s): 0< λ≤ 1, j= 1, … , J} and their derivatives whenα1,…, αJsatisfy a certain condition. This condition is satisfied ifα1,…, αJare algebraically independent over ℚ. More precisely, we establish a joint denseness result for values of those functions on vertical lines in the strip 1/2 < ℜs≤ 1. We also establish the functional independence of those functions in the sense of Voronin.