We survey the properties of sets of integers recognizable by automata when they are written in p-ary expansions. We focus on Cobham’s theorem which characterizes the sets recognizable in different bases p and on its generalization to N m due to Semenov. We detail the remarkable proof recently given by Muchnik for the theorem of Cobham-Semenov, the original proof being published in Russian. 1

We give algebraic proofs of transcendence over Q(X) of formal power series with rational coefficients, by using inter alia reduction modulo prime numbers, and the Christol theorem. Applications to generating series of languages and combinatorial objects are given. Keywords: transcendental formal power series, binomial series, automatic sequences, p-Lucas sequences, Chomsky-Schutzenberger theorem. 1 Introduction Formal power series with integer coefficients often occur as generating series. Suppose that a set E contains exactly a n elements of "size" n for each integer n: the generating series of this set is the formal power series P n0 a n X n (this series belongs to Z[[X]] ae Q[[X]]). Properties of this formal power series reflect properties of its coefficients, and hence properties of the set E. Roughly speaking, algebraicity of the series over Q(X) means that the set has a strong structure. For example, the Chomsky-Schutzenberger theorem [16] asserts that the generating seri...

dedicated to john tate on his 70th birthday We give another proof of Voloch's result on transcendence of the period of the Tate elliptic curve. The proof is based on the transcendence criterion of Christol involving notions of recognizable sequences and automata. 1996 Academic Press, Inc. Let p be a prime number, k be an algebraic closure of F p. Let q be a variable and consider a 4, a 6 # Z q (sometimes we consider them in k q) defined by a 4:=a 4(q):=: n 1 &5n 3 q n 1&q n, a 6:=a 6(q): =: n 1 &(7n 5 +5n 3) q n 12(1&q n) Let K:=k(a 4, a 6). In [V], Voloch proved the following function field analogue of the classical results of Siegel and Schneider proving the transcendence of periods of elliptic curves defined over algebraic number field. Theorem. The period q of the Tate elliptic curve y 2 +xy=x 3 +a 4x+a 6 over K is transcendental over K For more on the Tate curve (which we will not use directly) and for standard facts on modular forms (which we will use later), see [S, Chap. 5; M. Chap. 1], respectively. Voloch's nice proof involved approximating q by algebraic quantities and getting a contradiction by analyzing the Galois action using Igusa's theorem. We offer below a proof based on the criterion of algebraicity due to Christol. In [V] Voloch also proves transcendence of parameters of algebraic points by his method. It is unlikely that our method yields this easily.

Abstract. Denote by λ(n) Liouville’s function concerning the parity of the number of prime divisors of n. Using a theorem of Allouche, Mendès France, and Peyrière and many classical results from the theory of the distribution of prime numbers, we prove that λ(n) is not k–automatic for any k> 2. This yields that P∞ n=1 λ(n)Xn ∈ Fp[[X]] is transcendental over Fp(X) for any prime p> 2. Similar results are proven (or reproven) for many common number–theoretic functions, including ϕ, µ, Ω, ω, ρ, and others.

Abstract. The purpose of this expository article is to explain diverse new tools that automata theory provides to tackle transcendence problems in function field arithmetic. We collect and explain various useful results scattered in computer science, formal languages, logic literature and explain how they can be fruitfully used in number theory, dealing with transcendence, refined transcendence and classification problems.