Extention of the Laurinčikas — Matsumoto theorem ; Расширение теоремы Лауринчикаса — Матсумото

In 1975, S. M. Voronin discovered the remarkable universality property of the Riemann zeta-function 𝜁(𝑠). He proved that analytic functions from a wide class can be approximated with a given accuracy by shifts 𝜁(𝑠 + 𝑖𝜏 ), 𝜏 ∈ R, of one and the same function 𝜁(𝑠). The Voronin discovery inspired to continue investigations in the field. It turned out that some other zeta and 𝐿−functions as well as certain classes of Dirichlet series are universal in the Voronin sense. Among them, Dirichlet 𝐿-functions, Dedekind, Hurwitz and Lerch zeta-functions. In 2001, A. Laurinˇcikas and K. Matsumoto obtained the universality of zeta-functions 𝜁(𝑠, 𝐹) attached to certain cusp forms 𝐹. In the paper, the extention of the Laurinˇcikas-Matsumoto theorem is given by using the shifts 𝜁(𝑠 + 𝑖𝜙(𝜏 ), 𝐹) for the approximation of analytic functions. Here 𝜙(𝜏 ) is a differentiable real-valued positive increasing function, having, for 𝜏 > 𝜏0, the monotonic continuous positive derivative, satisfying, for 𝜏 → ∞, the conditions 1 𝜙′(𝜏) = 𝑜(𝜏 ) and 𝜙(2𝜏 ) max𝜏6𝑡62𝜏 1 𝜙′(𝑡) ≪ 𝜏 . More precisely, in the paper it is proved that, if 𝜅 is the weight of the cusp form 𝐹, 𝐾 is the compact subset of the strip {︀𝑠 ∈ C : 𝜅 2 < 𝜎 < 𝜅+1 2}︀ with connected complement, and 𝑓(𝑠) is a continuous non-vanishing function on 𝐾 which is analytic in the interior of 𝐾, then , for every 𝜀 > 0, the set {𝜏 ∈ R : sup𝑠∈𝐾 |𝜁(𝑠 + 𝑖𝜙(𝜏 ), 𝐹) − 𝑓(𝑠)| < 𝜀} has a positive lower density.