Joint discrete universality for Lerch zeta-functions / ; Совместная дискретная универсальность дзета-функций Лерха.

After Voronin’s work of 1975, it is known that some of zeta and 𝐿-functions are universal in the sense that their shifts approximate a wide class of analytic functions. Two cases of shifts, continuous and discrete, are considered. The present paper is devoted to the universality of Lerch zeta-functions 𝐿(𝜆, 𝛼, 𝑠), 𝑠 = 𝜎+𝑖𝑡, which are defined, for 𝜎 > 1, by the Dirichlet series with terms 𝑒2𝜋𝑖𝜆𝑚(𝑚+𝛼)−𝑠 with parameters 𝜆 ∈ R and 𝛼, 0 < 𝛼 6 1, and by analytic continuation elsewhere. We obtain joint discrete universality theorems for Lerch zeta-functions. More precisely, a collection of analytic functions 𝑓1(𝑠), . . . , 𝑓𝑟(𝑠) simultaneously is approximated by shifts 𝐿(𝜆1, 𝛼1, 𝑠+𝑖𝑘ℎ), . . . ,𝐿(𝜆𝑟, 𝛼𝑟, 𝑠+𝑖𝑘ℎ), 𝑘 = 0, 1, 2, . . . , where ℎ > 0 is a fixed number. For this, the linear independence over the field of rational numbers for the set {︀(log(𝑚 + 𝛼𝑗) : 𝑚 ∈ N0, 𝑗 = 1, . . . , 𝑟), 2𝜋ℎ}︀ is required. For the proof, probabilistic limit theorems on the weak convergence of probability measures in the space of analytic function are applied.