A Modification of the Mixed Joint Universality Theorem for a Class of Zeta Functions

The property of zeta functions on mixed joint universality in the Voronin's sense states that any two holomorphic functions can be approximated simultaneously with an accuracy of epsilon > 0 by suitable vertical shifts of the pair consisting the Riemann and Hurwitz zeta functions. A rather general result can be obtained for the classes of zeta functions, particularly when an approximating pair is composed of the Matsumoto zeta functions' class and the periodic Hurwitz zeta function. In this paper, we prove that this set of shifts has a strict positive density for all but at most countably epsilon > 0. Moreover, we provide concluding remarks on certain more general mixed tuples of zeta functions.