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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 $k\\ge1$.\n  A key feature of these functions is that their Dirichlet series admit a factorization into complex powers of the Riemann zeta function. 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