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

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(α_j, λ, s): 0 < λ ≤ 1, j = 1, … , J} and their derivatives when α₁,…, α_J satisfy a certain condition. This condition is satisfied if α₁,…, α_J 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.