Mittag-Leffler type theorems for Helson zeta-functions

Abstract

Let $f$ be a zero-free analytic function on $\Re(s) \geq 1$. We prove that there exists an entire zero-free function $g$ and a Helson zeta-function $\zeta_\chi(s)=\sum_{n=1}^\infty \chi(n) n^{-s}$, where $\chi(n)$ is a completely multiplicative unimodular function such that $f(s)=g(s) \zeta_\chi(s)$ for $\Re(s)>1$. By the Mittag-Leffler theorem this implies that a Helson zeta-function may have meromorphic continuation from $\Re(s)>1$ to the complex plane with a prescribed set of zeros and poles in the half plane $\Re(s)<1$. This improves on results of Seip and Bochkov-Romanov who proved the same result in the strip $21/40<\Re(s)<1$ and conditional on the Riemann hypothesis in the strip $1/2< \Re(s)<1$. 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 $\Re(s) >1$, 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.