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 on 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 $q$-automatic sequences or $q$-regular sequences.

翻译：生成级数在枚举组合学、解析组合学以及词上的组合学中至关重要。尽管初看之下，狄利克雷生成级数在这些领域中的应用似乎不及常生成级数和指数生成级数广泛，但已有大量重要论文揭示其核心作用，这在弗拉若莱及其多位合著者的工作中尤为显著。本文研究了在整数基$b$下缺失数字或数字块的整数的狄利克雷级数，即求和范围限定于那些展开式形成了字母表$\{0, 1, \dots, b-1\}$上不以$0$开头的所有词集合的严格子语言的整数。我们说明了如何统一并拓展纳桑森在2021年以及克勒与斯皮尔克在2009年所证明的结果。在此过程中，我们遇到了斯隆在线整数序列百科全书中的若干序列，以及一些著名的$q$-自动序列或$q$-正则序列。