Esthetic Numbers and Lifting Restrictions on the Analysis of Summatory Functions of Regular Sequences

TL;DR

This paper advances the asymptotic analysis of regular sequences by removing eigenvalue restrictions, enabling precise formulas for summatory functions, exemplified by counting esthetic numbers.

Contribution

It provides a new proof based on generating functions that lifts previous technical restrictions on eigenvalues in asymptotic analysis of regular sequences.

Findings

Derived precise asymptotic formulas for esthetic numbers.

Extended the analysis to all eigenvalue cases without restrictions.

Improved understanding of fluctuations in summatory functions.

Abstract

When asymptotically analysing the summatory function of a $q$-regular sequence in the sense of Allouche and Shallit, the eigenvalues of the sum of matrices of the linear representation of the sequence determine the "shape" (in particular the growth) of the asymptotic formula. Existing general results for determining the precise behavior (including the Fourier coefficients of the appearing fluctuations) have previously been restricted by a technical condition on these eigenvalues. The aim of this work is to lift these restrictions by providing a insightful proof based on generating functions for the main pseudo Tauberian theorem for all cases simultaneously. (This theorem is the key ingredient for overcoming convergence problems in Mellin--Perron summation in the asymptotic analysis.) One example is discussed in more detail: A precise asymptotic formula for the amount of esthetic numbers in the first~$N$ natural numbers is presented. Prior to this only the asymptotic amount of these numbers with a given digit-length was known.