On zeroes and poles of Helson zeta functions

TL;DR

This paper demonstrates that Helson zeta functions can have essentially arbitrary zeros and poles in certain regions of the complex plane, with results depending on the Riemann Hypothesis.

Contribution

It shows the flexibility of the zero and pole structure of Helson zeta functions in specified regions, unconditionally and under RH.

Findings

Arbitrary poles and zeros can occur in the strip 21/40 < Re s < 1 unconditionally.

Under RH, arbitrary poles and zeros can occur in the entire critical strip 1/2 < Re s < 1.

The results extend understanding of the analytic structure of Helson zeta functions.

Abstract

We show that the analytic continuations of Helson zeta functions $ \zeta_\chi (s)= \sum_1^{\infty}\chi(n)n^{-s} $ can have essentially arbitrary poles and zeroes in the strip $ 21/40 < \Re s < 1 $ (unconditionally), and in the whole critical strip $ 1/2 < \Re s <1 $ under Riemann Hypothesis.