Matrix methods for arithmetic functions

TL;DR

The paper presents a matrix-centric approach to arithmetic functions tied by additive convolution, showing how companion matrices built from divisor-sum and partition data yield determinant and partition-sum expressions for key sequences like $\tau(n+1)$ and $p(n)$. It develops a rigorous lemma-based framework linking the companion sequences via $J_n(\overline{j})$ and $H_n(\overline{h})$, and derives a generating-function perspective that expresses arithmetic-function values as partitions of determinants. It then explores deformations of the companion matrices, uncovers intriguing spectral phenomena (including apparent periodic envelopes in certain deformations) and validates these observations with extensive numerical plots, including a geometric survey of root-sets through color-coded visualizations. The work connects Ramanujan’s tau, the partition function, and modular-form coefficients within a unified determinant- and partition-based framework, and opens avenues for studying zero distributions and spectral geometry of arithmetic-function–driven matrices.

Abstract

We apply matrix methods to arithmetic functions by associating matrices to the functions in a manner drawn from the theory of symmetric functions. Then we study the characteristic polynomials of the associated matrices.

Figures (9)

• Figure 1: Minimum moduli of the roots of the $\chi_{\Pi^{(1)}_n}$
• Figure 2: Logarithms of the minimum moduli of the roots of the $\chi_{\Pi^{(1)}_n}$ (deformed matrices with $c=1$.)
• Figure 3: Maximum moduli of the roots of $\chi_{\Pi^{(1)}_n}$ (deformed matrices with $c=1$.)
• Figure 4: Minimum moduli of the roots of undeformed $\chi_{\Pi_n}$
• Figure 5: Maximum moduli of the roots of undeformed $\chi_{\Pi_n}$

Theorems & Definitions (23)

• Lemma 2.1: corrected version of Adams A22, entry 6.360
• proof
• Lemma 2.2
• proof
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Theorem 3.1