Helson zeta functions for characters with finitely many values

TL;DR

This paper demonstrates that Helson zeta functions associated with characters having finitely many values can have essentially arbitrary poles and zeros in specific regions of the complex plane, under certain conditions and hypotheses.

Contribution

It shows the flexibility in the pole-zero structure of Helson zeta functions for characters with finitely many values, including unconditional and conditional results based on the Riemann Hypothesis.

Findings

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

Under Riemann Hypothesis, arbitrary poles and zeros can be realized in the entire critical strip 1/2 < Re s < 1.

Symmetric sets of poles and zeros can be achieved with characters taking values ±1.

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 $.