On the Mishou Theorem for Zeta-Functions with Periodic Coefficients

Let a = { a m } and b = { b m } be two periodic sequences of complex numbers, and, additionally, a is multiplicative. In this paper, the joint approximation of a pair of analytic functions by shifts ( ζ n T ( s + i τ ; a ) , ζ n T ( s + i τ , α ; b ) ) of absolutely convergent Dirichlet series ζ n T ( s ; a ) and ζ n T ( s , α ; b ) involving the sequences a and b is considered. Here, n T → ∞ and n T ≪ T 2 as T → ∞ . The coefficients of these series tend to a m and b m , respectively. It is proved that the set of the above shifts in the interval [ 0 , T ] has a positive density. This generalizes and extends the Mishou joint universality theorem for the Riemann and Hurwitz zeta-functions.