Weighted Möbius Scores: A Unified Framework

• Weighted Möbius Scores are a unifying framework that extends classical Möbius inversion by incorporating problem-specific weights to quantify interactions in arithmetic, combinatorics, and geometry.
• They generalize summatory functions like the Mertens function and enable precise attributions in cooperative game theory and model interpretability via weighted arithmetic sums.
• Applications span analytic number theory, combinatorial feature attribution, and geometric topology, providing efficient and linearly decomposable measures of complexity and importance.

Weighted Möbius Scores provide a unifying mathematical framework in both analytic number theory and applied domains such as cooperative game theory, feature attribution in machine learning, and geometric analysis. At their core, these scores are based on weighting Möbius transforms—classical inversion operators on partially ordered sets—by problem-specific kernels or combinatorial coefficients, yielding measures of interaction, complexity, or importance. The abstract notion of “weighted Möbius score” therefore encompasses explicit arithmetic sums, combinatorial attributions, and geometric invariants, each formalized through domain-specific specialization of the weighting function or operation.

1. Foundational Definitions and Arithmetic Weighted Möbius Sums

The classical Möbius function $\mu(n)$, defined multiplicatively by

$\mu(1)=1, \qquad \mu(n)=(-1)^{\omega(n)} ~~\text{if %%%%1%%%% is square-free}, \qquad \mu(n)=0 ~~\text{otherwise}$

(where $\omega(n)$ is the count of distinct prime divisors), is central to analytic number theory. Its summatory function, the Mertens function $M(x)=\sum_{n\le x} \mu(n)$, exhibits deep cancellations and quasi-random oscillations across large intervals. Weighted Möbius sums generalize $M(x)$ by incorporating test functions, yielding

$W(x;y)=\sum_{n\le x}\mu(n)\,y(x/n)$

for suitable weights $y$. Notable choices include $y(t)=\log t$ (yielding the logarithmic sum $L(x)=\sum_{n\le x}\mu(n)\log(x/n)$), fractional-part weights for Cesàro means, and higher Bernoulli-polynomial kernels for generalized averages (Balazard, 2012). These yield a hierarchy of “weighted Möbius scores”, each encoding distinct arithmetic or analytic information, and all recoverable by Möbius inversion of the relevant transform (Schmidt, 2021, Balazard, 2012).

2. Möbius and Weighted Möbius Transformations in Combinatorics and Feature Attribution

On the Boolean lattice $2^{\mathcal D}$ (collections of all subsets of a $d$-element feature set), the Möbius transform

$$(\mu A)(S) = \sum_{T \subseteq S} (-1)^{|S| - |T|}\,A(T)$$

assigns to any valuation $A$ its set function “derivative”, isolating pure contributions of each subset. Weighted Möbius Scores (WMS) generalize this by specifying a weight function $\mathbf w(S, T)$, leading to

$$A_{\mathbf w}(S) = \sum_{T \subseteq \mathcal D} \mathbf w(S, T)\,A_{\mu(f)}(T)$$

where $A_{\mu(f)}(T)$ is the interaction score (Harsanyi dividend) for $T$ (Jiang et al., 2023). This unifies a plethora of attribution methods, including the Shapley value, Shapley interaction indices, and new attributions for higher-order feature interactions, mediation effects, and chain-of-thought prompt decompositions.

Key properties include:

• Linearity: $A_{\mathbf w}$ depends linearly on $A_{\mu(f)}$.
• Efficiency: The total attribution is $f(x)-f(x_{\setminus\mathcal D})$ if weights sum to one over $S$.
• Identifiability: Faithful $\mathbf w$ restricts attribution to relevant features.
• All linear, faithful attributions correspond to WMS for some $\mathbf w$ (Jiang et al., 2023).

3. Exact Arithmetic Formulas and Analytic Properties

Number-theoretic identities express the Mertens function $M(x)$ exactly in terms of weighted sums with combinatorial/auxiliary functions: $$M(x) = G(x) + \sum_{k \leq x} |g(k)|\,\pi(\lfloor x/k \rfloor)\,\lambda(k)$$ with $$G(x) = \sum_{n\le x} g(n) = \sum_{n\le x} \lambda(n) |g(n)|$$ and $g(n)$ the Dirichlet inverse of $\omega(n)+1$. The weights $|g(n)|$ (and combinatorially, $C_{\Omega}(n)$) display log-normal fluctuations, and their logarithms are asymptotically normal as $x\to\infty$ (Schmidt, 2021). Precise Dirichlet generating functions (DGFs) and analytic continuation arguments identify the growth and oscillation rates for these weighted sums. Explicit convolution formulas connect partial sums of $g(n)$, the prime counting function, and new exact rearrangements for $M(x)$, establishing the analytic depth of arithmetic weighted Möbius scores (Schmidt, 2021).

4. Generalizations: Weighted Möbius Scores in Structured Hierarchies

Weighted Möbius inversion extends naturally to partially ordered sets (posets), lattices, and directed acyclic multigraphs (DAMGs). In such structures, the incidence algebra supports a Möbius function $\mu_G$ and an associated Möbius transform: $$w(y) = v(y) - \sum_{x\in \mathrm{Anc}(y),\,x<y} w(x) = \sum_{x \le y} \mu_G(x, y) v(x).$$ Weighted Möbius scores on a DAMG project higher-order synergies to roots by weighted averaging over path kernels: $\mathrm{Sh}_r(v) = \sum_{y\in V} s(r|y)\,w(y)$ with $s(r|y)$ determined by “root strengths” and path weights. Characterizing axioms such as linearity, efficiency, weak-element removal, flat-hierarchy distribution, and localized symmetry uniquely determine these weighted Möbius scores, which generalize Shapley values and enable rigorous decomposition of contributions in hierarchical structures (Forré et al., 7 Oct 2025).

5. Möbius-Invariant Metrics and Weighted Scores in Geometric Topology

The term “weighted Möbius score” also appears in the context of differential geometry and knot theory, describing Möbius-invariant, parametrization-independent weighted inner products on the tangent bundle of the knot space. For a knot $f:S^1\to\mathbb{R}^3$ and weight function $\omega(f, t)$ satisfying

$\omega(T\circ f, t) = |T'(f(t))|^{-6} \omega(f, t)$

under Möbius transformations $T$, the bilinear form

$$\langle u, v \rangle_{\omega, f} = \int_{S^1} \omega(f, t)\,[u(t) \cdot v(t)]\,|f'(t)|dt$$

defines an invariant metric. Canonical choices for $\omega$ arise from Möbius energy densities $V(f, t)$ and general conformal invariants, yielding inner products and gradient flows on knot space that respect the full Möbius group (O'Hara, 2019). These constructions underlie geometric flows and shape analysis in conformal knot theory.

6. Inequalities, Asymptotics, and Quantitative Control

Weighted Möbius scores also encode analytic inequalities relating weighted summatory functions to the growth of $M(x)$. Classical results, such as

$$|m_1(x)| = \left| \frac{1}{x}\int_1^x M(t)\,dt \right| \leq \frac{1}{x} \int_1^x |M(t)|\,dt + \frac{1}{x}$$

and bounds for the normalized logarithmic sum $|m(x)| \leq 2 + \gamma$, provide explicit control of average behaviors in terms of the supremum of $|M(x)/x|$ (Balazard, 2012). These bounds sharpen under conjectural assumptions (e.g., the Riemann hypothesis yields $m_1(x)=O(x^{-1/2+\epsilon})$), and generalized Macleod identities provide a whole hierarchy of bounds for higher-order Cesàro and Riesz averages.

7. Applications and Interpretations across Disciplines

The Weighted Möbius Score paradigm links analytic number theory, combinatorics, cooperative game theory, machine learning interpretability, causal mediation, and differential geometry:

• In arithmetic, it decomposes $M(x)$ into weighted Liouville sums, providing a new lens for oscillatory phenomena and probabilistic modeling (Schmidt, 2021).
• In machine learning, it unifies model attribution, enabling the extraction of Harsanyi dividends, Shapley and Taylor indices, and indirect or mediated causal effects, all structured by linear weighting on Möbius coefficients (Jiang et al., 2023).
• In combinatorics and vector-valued valuations, it generalizes classical Shapley values to arbitrary weighted hierarchies, preserving essential axioms and yielding efficient algorithms (Forré et al., 7 Oct 2025).
• In geometric analysis, it provides the machinery for Möbius-invariant gradient flows and metric measurements on knot spaces (O'Hara, 2019).

This multifaceted structure positions Weighted Möbius Scores as a fundamental mathematical instrument for decomposing, weighting, and interpreting complex interactions and summations across discrete, algebraic, and geometric domains.