Abstract
For 0 < α ≤ 1 and 0 < λ ≤ 1, let L(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 when α1,…, αJ satisfy a certain condition. This condition is satisfied if α1,…, αJ are 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.
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Dedicated to Professors Antanas Laurinčikas and Eugenijus Manstavičius on the occasion of their 70th anniversary
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∗ This work was supported by JSPS KAKENHI grant No. 17K05160.
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Nagoshi, H. On a certain set of Lerch’s zeta-functions and their derivatives∗. Lith Math J 59, 111–130 (2019). https://doi.org/10.1007/s10986-019-09433-0
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DOI: https://doi.org/10.1007/s10986-019-09433-0