PRH | Inter | 5.2 • An Implementable Phase-Selection Algorithm for Helson Zeta

Since our first stop in RH quest was Helson zeta, here we complete that side of the story. A Helson zeta has unimodular, completely multiplicative coefficients: \[
\zeta_{\chi}(s)=\sum_{n\ge1}\chi(n)n^{-s}
=\prod_{p}(1-\chi(p)p^{-s})^{-1}\qquad(\Re s>1),
\] where $\chi:\mathbb{N}\to\mathbb{T}$ is completely multiplicative, $|\chi(n)|=1$. Andersson (2024) proved (and we reproved) that any prescribed discrete zero/pole multiset in the half-plane $\Re s<1$ (with mild local restrictions) is realized by a Helson zeta. We give a software-ready construction: a global, all-at-once optimization on the prime torus with exact analytic gradients,
run under a continuation schedule in the prime cutoff and the real part, together with short rigorous guardrails (uniform short-block linearization, grid$\to$sup upgrade, summable-error schedule, and Rouché/Hurwitz zero transfer). The method yields a completely multiplicative $\chi$ for which $\zeta_\chi$ matches any target zero pattern on expanding windows. We also show how to enforce pole-freeness at $s=1$ and (optionally) place all zeros on $\Re s=\tfrac12$.