Linear Combinations of ζ(s)/Πs Over Fq(x), where 1 ≤ s ≤ q − 2

AbstractCarlitz defined both a function ζ and a formal power series Π over Fq, analogous to the Riemann function ζ and to the real number π. Yu used Drinfeld modules to show the fraction ζ(s)/Πs is transcendental over Fq(x), when s is an integer not divisible by q − 1. In this paper we use the automata theory and Christol, Kamae, Mendes France and Rauzy theorem to prove the linear independence over Fq(x) of the fraction ζ(s)/Πs, for all integers s in [1, q − 2]