Transcendence of $L(1,\chi _s)/\Pi $ in positive characteristic. A simple automata-style proof

Abstract

For the field of formal Laurent series over a finite field, L. Carlitz defined $\Pi $, an analog of the real number $\pi $, and D. Goss defined $L(s,\chi )$, analogs of Dirichlet $L$-functions. G. Damamme proved in 1999 the transcendence of $L(1,\chi _s)/\Pi $ via a criterion of de Mathan. Then Y. Hu gave in 2018 an automata-style proof of the above result. In this work, we present another and much simpler automata-style proof.