Lithuanian Mathematical Journal

, Volume 59, Issue 1, pp 111–130 | Cite as

On a certain set of Lerch’s zeta-functions and their derivatives

  • Hirofumi NagoshiEmail author


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.


Lerch zeta-function joint value-distribution algebraic differential independence functional independence 


11M35 11J99 



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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Science and TechnologyGunma UniversityKiryuJapan

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