| Abstract [eng] |
For 𝑗=1,…,𝑟, let 𝒫𝑗 be a system of generalized prime numbers, 𝒩𝒫𝑗 the corresponding system of generalized integers, and 𝜁𝒫𝑗(𝑠), 𝑠=𝜎+𝑖𝑡, the Beurling zeta-function. In the paper, we consider simultaneous approximation of a collection of analytic functions by shifts (𝜁𝒫1(𝑠1+𝑖𝜏),…,𝜁𝒫𝑟(𝑠𝑟+𝑖𝜏)). For this, we require that the summatory functions satisfy the bounds ℳ𝒫𝑗(𝑥)−𝑎𝑗𝑥≪𝒫𝑗𝑥𝜃𝑗 with 𝑎𝑗>0 and 0⩽𝜃𝑗<1, 𝑗=1,…,𝑟. Moreover, we suppose that the mean square of 𝜁𝒫𝑗(𝑠) is bounded for 𝜎∈(𝜎𝒫𝑗,1) with some 𝜎𝒫𝑗 depending on 𝒫𝑗 and 𝜃𝑗. Then the main result of the paper asserts that there exists a closed non-empty set ℱ𝒫1,…,𝒫𝑟 of analytic functions such that its functions (𝑓1(𝑠1),…,𝑓𝑟(𝑠𝑟)) are approximated simultaneously by the above shifts of Beurling zeta-functions. It is proved that the set of approximating shifts has a positive lower density (or density with at most countably many exceptions of approximation accuracy). This shows a good joint approximation of Beurling zeta-functions. |
