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Generating series are crucial in enumerative combinatorics, analyticcombinatorics, and combinatorics on words. Though it might seem at first viewthat generating Dirichlet series are less used in these fields than ordinaryand exponential generating series, there are many notable papers where theyplay a fundamental role, as can be seen in particular in the work of Flajoletand several of his co-authors. In this paper, we study Dirichlet series ofintegers with missing digits or blocks of digits in some integer base $b$,i.e., where the summation ranges over the integers whose expansions form somelanguage 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 extendresults proved by Nathanson in 2021 and by K\"ohler and Spilker in 2009. Enroute, we encounter several sequences from Sloane's On-Line Encyclopedia ofInteger Sequences, as well as some famous $q$-automatic sequences or$q$-regular sequences.

Combinatorics on words and generating Dirichlet series of automatic sequences