Helson zeta functions for characters with finitely many values

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 for the function $\chi$ taking values in cubic roots of unity. If the sets are symmetric with respect to the real axis, the same can be achieved with $\chi$ taking values $\pm 1$.

arXiv admin note: substantial text overlap with arXiv:2106.15949