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.

翻译：生成级数在枚举组合学、解析组合学以及组合词学中具有核心地位。尽管初看起来狄利克雷生成级数在这些领域中的应用不如普通生成级数与指数生成级数广泛，但在许多重要文献中，它们发挥着基础性作用，这在Flajolet及其多位合作者的工作中尤为明显。本文研究在特定整数进制$b$下缺失数字或数字块的整数所对应的狄利克雷级数；即求和范围限定于那些在字母表$\{0, 1, \dots, b-1\}$上、不以$0$开头的全体字集之严格子语言所对应的整数展开式。我们展示了如何统一并推广Nathanson（2021年）与Köhler和Spilker（2009年）证明的结果。在此过程中，我们遇到了Sloane整数序列在线百科中的若干序列，以及一些著名的$b$-自动序列或$b$-正则序列。同时，我们也考察了一个非$b$-正则的具体序列。