Mittag-Leffler type theorems for Helson zeta-functions

Let f be a zero-free analytic function on (s) ≥ 1. We prove that there exists an entire zero-free function g and a Helson zeta-function ζ_χ(s)=∑ₙ₌₁^∞ χ(n) n⁻ˢ, where χ(n) is a completely multiplicative unimodular function such that f(s)=g(s) ζ_χ(s) for (s)>1. By the Mittag-Leffler theorem this implies that a Helson zeta-function may have meromorphic continuation from (s)>1 to the complex plane with a prescribed set of zeros and poles in the half plane (s)<1. This improves on results of Seip and Bochkov-Romanov who proved the same result in the strip 21/40<(s)<1 and conditional on the Riemann hypothesis in the strip 1/2< (s)<1. Our results also gives information on maximum domains of meromorphicity and analyticity of Helson zeta-functions and show that any open connected set U that includes the half plane (s) >1, may be a maximum domain of meromorphicity or of analyticity for a Helson zeta-function. This extends results of Bhowmik and Schlage-Puchta to Dirichlet series with Euler products.