Sur la transcendance de la série formelle \(\Pi\) . (On the transcendence of formal power series \(\Pi\) ). (French) Zbl 0709.11067

Allouche, Jean-Paul

Sémin. Théor. Nombres Bordx., Sér. II 2, No. 1, 103-117 (1990).

Let \({\mathbb{A}}={\mathbb{F}}_ q[x]\) where \({\mathbb{F}}_ q\) is the finite field with q elements. There is a very well-known analogy between the P.I.D. \({\mathbb{A}}\) and the usual integers \({\mathbb{Z}}\). This analogy even extends to the existence of special transcendental functions. Indeed, let R be either \({\mathbb{A}}\) or \({\mathbb{Z}}\) and let \({\mathbb{P}}\) be the subset of positives (for \({\mathbb{Z}})\) or monics (for \({\mathbb{A}})\). We formally put \(\zeta_ R(s)=\sum_{a\in {\mathbb{P}}}a^{-s};\) we consider \(\zeta_ R\) as a function on positive integers where it converges absolutely except for a pole at \(s=1\) for \({\mathbb{Z}}\). Moreover, considering R as a “lattice”, we are led to the construction of “exponential functions”; in both cases rationality considerations force the introduction of a constant “\(\pi\) ”.
Let \(\alpha\) be the number of units in R and call a positive integer “R- even” if it is divisible by \(\alpha\). Then one knows that \(\zeta_ R(i)/\pi^ i\) is in the quotient field of R for R-even integers. (This is due to Euler for \({\mathbb{Z}}\); for \({\mathbb{A}}\) it was shown by Carlitz in the 30’s and independently rediscovered by the present author in the 70’s.)
The number \(\pi\) is well-known to be transcendental when \(R={\mathbb{Z}}\). When \(R={\mathbb{A}}\), the result was first established by L. I. Wade. The purpose of the present article is to use the work of Christol, Kamae, Mendès France and Rauzy to give an elementary proof of this last theorem.

Reviewer: D.Goss

MSC:

11R58	Arithmetic theory of algebraic function fields
11J81	Transcendence (general theory)
11T55	Arithmetic theory of polynomial rings over finite fields
14G15	Finite ground fields in algebraic geometry

Keywords:

Carlitz module; transcendence of \(\pi \); Carlitz zeta function

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