On zeroes and poles of Helson zeta function

The structure of poles and zeroes of the Helson zeta function, $ \zeta_\chi (s)= \sum_1^{\infty}\chi(n)n^{-s} $, is studied. In particular, it is shown that two arbitrary disjoint sets in the critical strip $ 21/40 < \Re s < 1 $ not accumulating off the left boundary $ \Re s = 21/40 $ are the sets of zeroes and poles of $ \zeta_\chi $, respectively, for an appropriate choice of the completely multiplicative unimodular function $ \chi $. This is a joint work with I. Bochkov.