[Paper Review] The zeta function of a finite category which has Möbius inversion

Kazunori Noguchi|arXiv (Cornell University)|May 20, 2012

semigroups and automata theory5 references3 citations

TL;DR

This paper proves a conjecture linking the zeta function and Euler characteristic of a finite category with Möbius inversion. It establishes that the zeta function is a rational function whose singularities and zeros correspond exactly to the roots of the characteristic polynomial of the category's adjacency matrix, and verifies that the Euler characteristic equals a specific sum over residues of the zeta function's logarithmic derivative.

ABSTRACT

We prove certain conjecture holds true for a finite category which has Möbius inversion. The conjecture states a relationship between the zeta function of a finite category and the Euler characteristic of a finite category.

Motivation & Objective

• To verify a conjecture relating the zeta function and Euler characteristic of a finite category with Möbius inversion.
• To establish the functional form of the zeta function for such categories, showing it is a finite product of rational and exponential terms.
• To demonstrate that the Euler characteristic of the category equals a specific sum over residues of the logarithmic derivative of the zeta function.
• To characterize the singular points and zeros of the zeta function in terms of the eigenvalues of the adjacency matrix of the category.
• To prove that the sum of the inverse eigenvalues and certain coefficients in the zeta function expansion equals the series Euler characteristic.

Proposed method

• Define the zeta function of a finite category as an exponential generating function over the number of chains of morphisms of length m.
• Use the adjacency matrix $ A_I $ of the category to express the zeta function in terms of the determinant $ \det(E - A_I z) $, linking it to the characteristic polynomial.
• Apply residue calculus and partial fraction decomposition to the logarithmic derivative of the zeta function, expressing it as a sum over terms involving $ \frac{A_{k,j}}{(z - \alpha_k)^j} $.
• Use properties of the adjugate matrix $ \mathrm{adj}(A_I) $ and the trace of the inverse to relate the sum of residues to the Euler characteristic.
• Leverage the fact that the zeta function's singularities and zeros coincide with the roots of $ \det(E - A_I z) $, proving that each root $ \alpha_k $ corresponds to a pole or zero.
• Use induction and coefficient comparison in the Laurent series expansion to show that the sum of coefficients $ \sum \beta_i / \alpha_i + \sum (-1)^j \gamma_j / \delta_j^{j+1} $ equals the Euler characteristic.

Experimental results

Research questions

• RQ1 Does the zeta function of a finite category with Möbius inversion admit a finite product decomposition of the form $ \prod \frac{1}{(1 - \alpha_i z)^{\beta_i}} \exp\left(\sum \frac{\gamma_j z^j}{j(1 - \delta_j z)^j}\right) $?
• RQ2 Is the sum of the exponents $ \sum \beta_i $ in the zeta function equal to the number of objects in the category?
• RQ3 Are the parameters $ \alpha_i $ in the zeta function's product form eigenvalues of the adjacency matrix $ A_I $, and thus algebraic integers?
• RQ4 Does the sum $ \sum \frac{\beta_i}{\alpha_i} + \sum (-1)^j \frac{\gamma_j}{\delta_j^{j+1}} $ equal the series Euler characteristic $ \chi_\Sigma(I) $ of the category?
• RQ5 Are the singular points and zeros of the zeta function precisely the roots of the characteristic polynomial $ \det(E - A_I z) $?

Key findings

• The zeta function of a finite category with Möbius inversion is a finite product of rational and exponential terms, confirming Conjecture 1.1 (C1).
• The sum of the exponents $ \sum \beta_i $ in the zeta function equals the number of objects in the category, verifying Conjecture 1.1 (C2).
• Each $ \alpha_i $ in the zeta function's product form is an eigenvalue of the adjacency matrix $ A_I $, and hence an algebraic integer, confirming Conjecture 1.1 (C3).
• The sum $ \sum \frac{\beta_i}{\alpha_i} + \sum (-1)^j \frac{\gamma_j}{\delta_j^{j+1}} $ equals the series Euler characteristic $ \chi_\Sigma(I) $, confirming Conjecture 1.1 (C4).
• The singular points and zeros of the zeta function are exactly the roots of the characteristic polynomial $ \det(E - A_I z) $, as shown in Corollary 3.2.
• The sum of the entries of the inverse adjacency matrix $ \mathrm{sum}(A_I^{-1}) $ is equal to the sum of residues of the logarithmic derivative of the zeta function, derived via coefficient comparison in the Laurent expansion.