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 Möbius 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 $\{{\epsilon }_k{\}}_{k\ge 1}$ via the rule $\mathfrak{f}(p^k)={\epsilon }_k$ for every prime $p$ and every $k\ge 1$. 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 ${\sum }_{n\ge 1}\mathfrak{f}(n)e^{-n/x}$, 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 Möbius functions.