Combinatorics on words and generating Dirichlet series of automatic sequences

TL;DR

This work studies Dirichlet generating functions $F_L(z)$ for integers whose base-$b$ expansions lie in a language $L$, focusing on abscissae of convergence and how digit-block restrictions influence convergence. Using both Titchmarsh-type summatory-function arguments and language-counting techniques, it unifies and extends results of Köhler–Spilker (2009) and Nathanson (2021) for restricted Dirichlet series, deriving explicit abscissa formulas like $\Theta_{\mathcal D}=\frac{1}{\log b}\sum_{\ell=0}^{b-1} \alpha_\ell \log(b-\ell)$ and showing $r_0=\log_b \lambda$ with $\lambda$ the dominant root of a combinatorial polynomial. The paper also develops a general automatic-regular framework giving abscissae for $F_L(z)$ when the digit-sum sequence is $b$-automatic or $b$-regular, and it demonstrates non-regular cases and alternative proofs via summatory functions. Through numerous examples, including connections to Sloane’s OEIS, the authors highlight the broad applicability of these methods to block-avoidance languages and present open problems on pole structures and meromorphic continuations.

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 Köhler 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.