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Logic and precognizable sets of integers
 Bull. Belg. Math. Soc
, 1994
"... We survey the properties of sets of integers recognizable by automata when they are written in pary 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 ..."
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Cited by 88 (4 self)
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We survey the properties of sets of integers recognizable by automata when they are written in pary 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 CobhamSemenov, the original proof being published in Russian. 1
Transcendence of Formal Power Series With Rational Coefficients
, 1999
"... 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 po ..."
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Cited by 12 (2 self)
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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, pLucas sequences, ChomskySchutzenberger 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 ChomskySchutzenberger theorem [16] asserts that the generating seri...
(NON)AUTOMATICITY OF NUMBER THEORETIC FUNCTIONS
, 2008
"... 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 yiel ..."
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Cited by 4 (2 self)
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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.
Automatastyle proof of Voloch’s result on transcendence
 J. NUMBER THEORY
, 1996
"... 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. ..."
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Cited by 2 (1 self)
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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.
Automata Methods in Transcendence
"... 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 ho ..."
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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.
Some aspects of analytic number theory: Parity, . . .
, 2009
"... Questions on parities play a central role in analytic number theory. Properties of the partial sums of parities are intimate to both the prime number theorem and the Riemann hypothesis. This thesis focuses on investigations of Liouville’s parity function and related completely multiplicative parit ..."
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Questions on parities play a central role in analytic number theory. Properties of the partial sums of parities are intimate to both the prime number theorem and the Riemann hypothesis. This thesis focuses on investigations of Liouville’s parity function and related completely multiplicative parity functions. We give results about the partial sums of parities as well as transcendence of functions and numbers associated to parities. For example, we show that the generating function of Liouville’s parity function is transcendental over the ring of rational functions with coefficients from a finite field. Within the course of investigation, relationships to finite automata are also discussed.