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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2512.18936 |
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| _version_ | 1866917176940691456 |
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| author | Saraeb, Ali |
| author_facet | Saraeb, Ali |
| contents | We study \emph{unimodular fake} $μ's$, i.e. multiplicative functions $\mathfrak f: \N \to \mathbb{S}^1 \cup \{0\} $ determined by a fixed sequence $\{\varepsilon_k\}_{k\ge 0} \subset \mathbb{S}^1 \, \cup \, \{0\}$ via the rule $\mathfrak f(p^k)=\varepsilon_k$ for every prime $p$ and $k \ge 0$. The Dirichlet series of a given $\mathfrak f$ admits the Euler product \[ F_{\mathfrak f}(s)=\sum_{n\ge1}\frac{\mathfrak f(n)}{n^s} =\prod_p g_{\mathfrak f}(p^{-s}),\qquad g_{\mathfrak f}(u)=\sum_{k\ge0}\varepsilon_k u^k, \] and the 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 Simple Zeros Conjecture, we derive an explicit formula for $A_{\mathfrak f}^{\exp}(x)= \sum_{n \ge 1} \mathfrak f(n) \, e^{-n/x} $ of the form \[ A_{\mathfrak f}^{\exp}(x) -Δ_1(x;z,w) = Δ_{1/2}(x;z,w)\;+\;\sum_ρΔ_ρ(x;z,w,\mathfrak f)\;+\;\mathcal E(x). \] To our knowledge, our expansion is the first extension of the Selberg-Delange method for Dirichlet series of the form $ζ(s)^{\,z}\,ζ(2s)^{\,w}\,G(s)$ that, beyond the main term from $s=1$, also extracts lower-order contributions from the singularities on the critical line $\Re(s)=1/2$.
On the other hand, we introduce a notion of \emph{bias} at the natural scale $x^{1/2}(\Log x)^{w-1}$ and obtain an explicit criterion distinguishing \emph{persistent}, \emph{apparent}, and \emph{unbiased} behavior in this regime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_18936 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Unimodular Fake Mobius Functions Saraeb, Ali Number Theory We study \emph{unimodular fake} $μ's$, i.e. multiplicative functions $\mathfrak f: \N \to \mathbb{S}^1 \cup \{0\} $ determined by a fixed sequence $\{\varepsilon_k\}_{k\ge 0} \subset \mathbb{S}^1 \, \cup \, \{0\}$ via the rule $\mathfrak f(p^k)=\varepsilon_k$ for every prime $p$ and $k \ge 0$. The Dirichlet series of a given $\mathfrak f$ admits the Euler product \[ F_{\mathfrak f}(s)=\sum_{n\ge1}\frac{\mathfrak f(n)}{n^s} =\prod_p g_{\mathfrak f}(p^{-s}),\qquad g_{\mathfrak f}(u)=\sum_{k\ge0}\varepsilon_k u^k, \] and the 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 Simple Zeros Conjecture, we derive an explicit formula for $A_{\mathfrak f}^{\exp}(x)= \sum_{n \ge 1} \mathfrak f(n) \, e^{-n/x} $ of the form \[ A_{\mathfrak f}^{\exp}(x) -Δ_1(x;z,w) = Δ_{1/2}(x;z,w)\;+\;\sum_ρΔ_ρ(x;z,w,\mathfrak f)\;+\;\mathcal E(x). \] To our knowledge, our expansion is the first extension of the Selberg-Delange method for Dirichlet series of the form $ζ(s)^{\,z}\,ζ(2s)^{\,w}\,G(s)$ that, beyond the main term from $s=1$, also extracts lower-order contributions from the singularities on the critical line $\Re(s)=1/2$. On the other hand, we introduce a notion of \emph{bias} at the natural scale $x^{1/2}(\Log x)^{w-1}$ and obtain an explicit criterion distinguishing \emph{persistent}, \emph{apparent}, and \emph{unbiased} behavior in this regime. |
| title | Unimodular Fake Mobius Functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2512.18936 |