On the Mishou Theorem for Zeta-Functions with Periodic Coefficients

Abstract

Let a={am} and b={bm} 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 (ζnT(s+iτ;a),ζnT(s+iτ,α;b)) of absolutely convergent Dirichlet series ζnT(s;a) and ζnT(s,α;b) involving the sequences a and b is considered. Here, nT→∞ and nT≪T2 as T→∞. The coefficients of these series tend to am and bm, 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.

A Discrete Version of the Mishou Theorem Related to Periodic Zeta-Functions

“…We recall a theorem from [4] which extends the Mishou theorem for ζ u T (s; a) and ζ u T (s, α; b) with u T → ∞. For its statement, we need some notation and definitions.…”