Mittag-Leffler type theorems for Helson zeta-functions

TL;DR

This paper proves that Helson zeta-functions can be extended meromorphically with prescribed zeros and poles, improving previous results and providing new insights into their maximum domains of analyticity and meromorphicity.

Contribution

It establishes that Helson zeta-functions can have meromorphic continuations with arbitrary prescribed zeros and poles, extending previous results and covering broader domains.

Findings

Helson zeta-functions can be meromorphically continued with prescribed zeros and poles.

Any open set containing the half-plane Re(s)>1 can be a maximum domain of meromorphicity.

Results extend to Dirichlet series with Euler products.

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.