Extention of the Laurincikas - Matsumoto theorem
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 + ), ∈ 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 > 𝜏0, the monotonic continuous positive derivative, satisfying, for → ∞, the conditions 1 𝜙′(𝜏) = 𝑜(𝜏 ) and 𝜙(2𝜏 ) max𝜏6𝑡62𝜏 1 𝜙′(𝑡) ≪ . More precisely, in the paper it is proved that, if is the weight of the cusp form 𝐹, is the compact subset of the strip {︀ ∈ C : 2 < < 𝜅+1 2 }︀ with connected complement, and 𝑓(𝑠) is a continuous non-vanishing function on which is analytic in the interior of 𝐾, then , for every > 0, the set {𝜏 ∈ R : sup𝑠∈𝐾 |𝜁(𝑠 + 𝑖𝜙(𝜏 ), 𝐹) - 𝑓(𝑠)| < 𝜀} has a positive lower density.