Unimodular Fake Mobius Functions

Abstract

Let $\mathbb{S}^1$ denote the unit circle. We introduce and develop the analytic and bias theory of unimodular fake M\"obius functions, i.e. multiplicative functions $\mathfrak{f}:\mathbb{N} \to \mathbb{S}^1 \cup \{0\}$ whose prime-power values are prescribed by a fixed sequence $\{\varepsilon_k\}_{k\ge1}$ via the rule $\mathfrak{f}(p^k)=\varepsilon_k$ for every prime $p$ and every $k\ge1$. A key feature of these functions is that their Dirichlet series admit a factorization into complex powers of the Riemann zeta function. Our main analytic result is an explicit formula for the smoothed summatory function $\sum_{n\ge1}\mathfrak{f}(n)e^{-n/x}$, consisting of a leading main term together with a sequence of lower-order terms. The formula may be viewed as an extension of the Selberg-Delange method and is expected to be of independent interest. As an application, we introduce a notion of bias at a natural scale and obtain an explicit criterion distinguishing persistent bias, apparent bias, and no bias for unimodular fake M\"obius functions.