Abstract :
[en] 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.
