Fake Mobius-type Functions on the Unit Circle
Ali Saraeb
TL;DR
The paper extends the fake Möbius framework by allowing complex unit-circle prime-power data and derives a universal zeta-factorization F_f(s)=ζ(s)^z ζ(2s)^w G_f(s). It develops a rigorous analytic apparatus—complex-power bounds for ζ(s), contour shifts with Hankel detours, and Perron-type smoothing—to obtain explicit Laplace-representations and Watson-type expansions for the smoothed summatory function A_f^{exp}(x). A central contribution is a precise bias analysis at the critical x^{1/2} (log x)^{w-1} scale, distinguishing persistent, apparent, and no-bias regimes under the RH/SZC framework and mild zero-spacing assumptions. The work clarifies how the leading singular data (z,w) governs the bias, while higher-order ε_k affect only lower-order terms, and sets the stage for future exploration of complementary parameter regions.
Abstract
We study \emph{fake Mobius-type functions on the unit circle}: multiplicative functions $\mathfrak f$ determined by prime-power data $\mathfrak f(p^k)=\varepsilon_k\in\mathbb S^1\cup\{0\}$, independent of $p$. Their Dirichlet series admits the Euler product \[ F_{\mathfrak f}(s)=\sum_{n\ge1}\frac{\mathfrak f(n)}{n^s} =\prod_p g(p^{-s}),\qquad g(u)=\sum_{k\ge0}\varepsilon_k u^k, \] and a canonical zeta-factorization \[ F_{\mathfrak f}(s)=ζ(s)^{\,z}\,ζ(2s)^{\,w}\,G_{\mathfrak f}(s), \qquad z=\varepsilon_1,\ \ w=\varepsilon_2-\frac{\varepsilon_1(\varepsilon_1+1)}{2}, \] where $G_{\mathfrak f}(s)$ is a holomorphic Euler product on $\Re s>1/3$. Assuming the Riemann hypothesis and simplicity of zeros, we derive a general explicit and asymptotic formula for the summatory functions $A_{\mathfrak f}^{\exp}(x)$ of the form \[ A_{\mathfrak f}^{\exp}(x) -Δ_1(x;z,w) = Δ_{1/2}(x;z,w)\;+\;\sum_ρΔ_ρ(x;z,w,\mathfrak f)\;+\;\mathcal E(x), \] where $Δ_1(x;z,w)$ is the main term coming from the $s=1$ singularity of $ζ(s)^z$, $Δ_{1/2}(x;z,w)$ is the secondary term coming from the $s=1/2$ singularity of $ζ(2s)^w$, the terms $Δ_ρ(x;z,w,\mathfrak f)$ arise from the singularities of $ζ(s)^z$ at nontrivial zeros $ρ$ of $ζ(s)$, and $\mathcal E(x)$ is an error term. Furthermore, we introduce a notion of \emph{bias} at the natural scale $x^{1/2}(\Log x)^{w-1}$ and obtain an explicit criteria distinguishing persistent, apparent, and unbiased behavior in this regime. The present paper treats the major parameter region \[ -1\le\Re(z)\le1,\qquad -2\le\Re(w)<1, \] and we choose to defer the complementary ranges of $(z,w)$ to a future paper as a different analytic approach is required there.