On zeroes and poles of Helson zeta functions

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 $21\mathrm{/}40<\mathrm{\Re }s<1$ (unconditionally), and in the whole critical strip $1\mathrm{/}2<\mathrm{\Re }s<1$ under Riemann Hypothesis.