On zeroes and poles 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.

8 pages