A note on the relation between fixed point and orbit count sequences

TL;DR

This paper explores the mathematical relationship between fixed point and orbit count sequences using linear algebra on arithmetic functions, providing spectral, asymptotic analyses and explicit formulas involving Gaussian binomial coefficients.

Contribution

It introduces a novel linear framework to analyze fixed point and orbit sequences, deriving spectral and asymptotic properties and explicit formulas.

Findings

Spectral properties of the linear mappings are characterized.

Asymptotic behaviors of the sequences are derived.

Explicit formulas involve Gaussian binomial coefficients.

Abstract

The relation between fixed point and orbit count sequences is investigated from the point of view of linear mappings on the space of arithmetic functions. Spectral and asymptotic properties are derived and several quantities are explicitly given in terms of Gaussian binomial coefficients.