Dirichlet series associated with strongly \(q\)-multiplicative functions (Q1876049)

Dirichlet series associated with the Thue-Morse sequence have been studied by \textit{J.-P. Allouche} and \textit{H. Cohen} [Bull. Lond. Math. Soc. 17, 531--538 (1985; Zbl 0577.10036)]. The case of series associated with general automatic sequences has been adressed by \textit{J.-P. Allouche}, \textit{M. Mendès France} and \textit{J. Peyrière} [J. Number Theory 81, No. 2, 359--373 (2000; Zbl 0971.11006)]. In the interesting paper under review, the author studies Dirichlet series associated with strongly \(q\)-multiplicative sequences whose average on \([0,q-1]\) is zero. Such series are known to have an analytic continuation to the whole plane [\textit{P. J. Grabner}, Acta. Arith. 65, No. 1, 85--96 (1993; Zbl 0783.11035)]. The author gives a different proof that also permits to prove that the order of these Dirichlet series is equal to 2. He also succeeds in proving that most of the trivial zeros of these series are simple.