Combinatorial proof of the transcendence of $L(1,\chi_s)/\Pi$

TL;DR

This paper presents a combinatorial proof demonstrating the transcendence of a specific function ratio in characteristic p, leveraging Christol's theorem and properties of automatic sequences, offering an alternative to previous proofs.

Contribution

It provides a new combinatorial proof of the transcendence of $L(1, heta_s)/oldsymbol{ au}$, distinct from Damamme's analytic approach, using automata theory.

Findings

Proof confirms transcendence of $L(1, heta_s)/oldsymbol{ au}$ in characteristic p.

Utilizes Christol's theorem and properties of automatic sequences.

Offers a combinatorial alternative to existing transcendence proofs.

Abstract

We give a combinatorial proof of the transcendence of $L(1,\chi_s)/\Pi$, where $L(1,\chi_s)$ (resp. $\Pi$) is the analogue in characteristic $p$ of the function $L$ of Dirichlet (resp. $\pi$). This result has been proven by G. Damamme using the criteria of de Mathan. Our proof is based on the Theorem of Christol and another property of $k$-automatic sequences.