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 _ (s) =₍=₁^ (n) n^-s, where (n) is a completely multiplicative unimodular function such that f (s) =g (s) _ (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, 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.