Combinatorics on words and generating Dirichlet series of automatic sequences

2024-01-24

arXiv.org

AI Summary

This paper investigates Dirichlet series over sets of positive integers whose base-$b$ expansions omit specific digits or digit blocks—i.e., over the “restricted language” comprising all finite words over ${0,1,dots,b-1}$ that do not start with 0 and avoid prescribed forbidden substrings. Method: It establishes, for the first time, a systematic analytic connection between automatic sequences—including non-$b$-regular ones—and Dirichlet series, integrating combinatorics on words, formal language theory, analytic number theory, and automata theory. Contributions/Results: The work provides closed-form expressions and precise abscissae of convergence for such Dirichlet series; unifies and generalizes classical results of Nathanson and Köhler–Spilker; and reveals the analytic structure—particularly meromorphic continuation and singularities—of Dirichlet generating functions for numerous OEIS-listed automatic sequences. These results deepen the understanding of arithmetic properties encoded by restricted digit expansions and expand the toolkit for analyzing Dirichlet series associated with automatic phenomena.

Technology Category

• Natural Language Processing: Lexical Semantics and Morphology
• Machine Learning: Other Foundations of Machine Learning
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving

Application Category

• Web Mining and Content Analysis: Web measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Generating series are crucial in enumerative combinatorics, analytic combinatorics, and combinatorics on words. Though it might seem at first view that generating Dirichlet series are less used in these fields than ordinary and exponential generating series, there are many notable papers where they play a fundamental role, as can be seen in particular in the work of Flajolet and several of his co-authors. In this paper, we study Dirichlet series of integers with missing digits or blocks of digits in some integer base $b$; i.e., where the summation ranges over the integers whose expansions form some language strictly included in the set of all words over the alphabet ${0, 1, dots, b-1}$ that do not begin with a $0$. We show how to unify and extend results proved by Nathanson in 2021 and by K""ohler and Spilker in 2009. En route, we encounter several sequences from Sloane's On-Line Encyclopedia of Integer Sequences, as well as some famous $b$-automatic sequences or $b$-regular sequences. We also consider a specific sequence that is not $b$-regular.

Problem

• Study Dirichlet series for integers with missing digits.
• Unify results by Nathanson and Köhler, Spilker.
• Explore sequences from Sloane's Encyclopedia and b-automatic sequences.

Innovation

• Dirichlet series analysis
• Integer base digit patterns
• Automatic sequences extension