| Abstract [eng] |
Since 1975, it has been known that the Hurwitz zeta-function has a unique property to approximate by its shifts all analytic functions defined in the strip π={π =π+ππ‘:1/2<π<1}. However, such an approximation causes efficiency problems, and applying short intervals is one of the measures to make that approximation more effective. In this paper, we consider the simultaneous approximation of a tuple of analytic functions in the strip π by discrete shifts (π(π +ππβ1,πΌ1),β¦,π(π +ππβπ,πΌπ)) with positive β1,β¦,βπ of Hurwitz zeta-functions in the interval [π,π+π] with π=max1β©½πβ©½π(ββ1π(πβπ)23/70). Two cases are considered: 1β the set {(βπlog(π+πΌπ),πββ0,π=1,β¦,π),2π} is linearly independent over β; and 2β a general case, where πΌπ and βπ are arbitrary. In case 1β , we obtain that the set of approximating shifts has a positive lower density (and density) for every tuple of analytic functions. In case 2β , the set of approximated functions forms a certain closed set. For the proof, an approach based on new limit theorems on weakly convergent probability measures in the space of analytic functions in short intervals is applied. The power π=23/70 comes from a new mean square estimate for the Hurwitz zeta-function. |
