Unimodular Fake Mobius Functions

TL;DR

This paper investigates unimodular fake Möbius functions, deriving explicit formulas for their exponential sums under the Riemann hypothesis, extending the Selberg-Delange method to include lower-order critical line contributions, and analyzing bias phenomena.

Contribution

It introduces a novel extension of the Selberg-Delange method for specific Dirichlet series, capturing lower-order terms from the critical line and defining a bias concept at the natural scale.

Findings

Explicit formula for exponential sums of unimodular fake Möbius functions.

Extension of the Selberg-Delange method to include critical line contributions.

Classification of bias behavior in the natural scale regime.

Abstract

We study \emph{unimodular fake} $\mu's$, 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 \[ F_{\mathfrak f}(s)=\zeta(s)^{\,z}\,\zeta(2s)^{\,w}\,G_{\mathfrak f}(s), \qquad z=\varepsilon_1,\ \ w=\varepsilon_2-\frac{\varepsilon_1(\varepsilon_1+1)}{2}, \] 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 \[ A_{\mathfrak f}^{\exp}(x) -\Delta_1(x;z,w) = \Delta_{1/2}(x;z,w)\;+\;\sum_{\rho}\Delta_\rho(x;z,w,\mathfrak f)\;+\;\mathcal E(x). \] To our knowledge, our expansion is the first extension of the Selberg-Delange method for Dirichlet series of the form $\zeta(s)^{\,z}\,\zeta(2s)^{\,w}\,G(s)$ 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.