On Approximation of Analytic Functions by Periodic Hurwitz Zeta-Functions

摘要

The periodic Hurwitz zeta-function zeta(s, alpha; a), s = sigma + it, with parameter 0 alpha = 1 and periodic sequence of complex numbers a = {a(m)} is defined, for sigma 1, by the series Sigma(infinity)(m=0) a(m)/(m+alpha)(s), and can be continued moromorphically to the whole complex plane. It is known that the function zeta(s, alpha; a) with transcendental or rational alpha is universal, i.e., its shifts zeta(s+i tau, alpha; a) approximate all analytic functions defined in the strip D = {s is an element of C : 1/2 sigma 1}. In the paper, it is proved that, for all 0 alpha = 1 and a, there exists a non-empty closed set F-alpha,F-a of analytic functions on D such that every function f is an element of F-alpha,(a) can be approximated by shifts zeta(s + i tau, alpha; a).