Results 1 
5 of
5
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 ..."
Abstract

Cited by 68 (4 self)
 Add to MetaCart
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
Sur un problème de Gelfond: la somme des chiffres des nombres premiers
, 2010
"... In this article we answer a question proposed by Gelfond in 1968. We prove that the sum of digits of prime numbers written in a basis q> 2 is equidistributed in arithmetic progressions (except for some well known degenerate cases). We prove also that the sequence.˛sq.p/ / where p runs through the p ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
In this article we answer a question proposed by Gelfond in 1968. We prove that the sum of digits of prime numbers written in a basis q> 2 is equidistributed in arithmetic progressions (except for some well known degenerate cases). We prove also that the sequence.˛sq.p/ / where p runs through the prime numbers is equidistributed modulo 1 if and only if ˛ 2 � n �.
(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 yields ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
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.
Automaticity IV: Sequences, Sets, and Diversity
 J. Th'eorie Nombres Bordeaux
, 1996
"... This paper studies the descriptional complexity of (i) sequences over a finite alphabet; and (ii) subsets of N (the natural numbers). If (s(i)) i0 is a sequence over a finite alphabet \Delta, then we define the kautomaticity of s, A k s (n), to be the smallest possible number of states in any det ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
This paper studies the descriptional complexity of (i) sequences over a finite alphabet; and (ii) subsets of N (the natural numbers). If (s(i)) i0 is a sequence over a finite alphabet \Delta, then we define the kautomaticity of s, A k s (n), to be the smallest possible number of states in any deterministic finite automaton that, for all i with 0 i n, takes i expressed in basek as input and computes s(i). We give examples of sequences that have high automaticity in all bases k; for example, we show that the characteristic sequence of the primes has k automaticity A k s (n) = \Omega\Gamma n 1=43 ) for all k 2, thus making quantitative the classical theorem of Minsky and Papert that the set of primes expressed in base2 is not regular. We give examples of sequences with low automaticity in all bases k, and low automaticity in some bases and high in others. We also obtain bounds on the automaticity of certain sequences that are fixed points of homomorphisms, such as the Fibonac...
Number Theory And Formal Languages
 Emerging Applications of Number Theory, IMA Volumes in Mathematics and Applications
, 1999
"... . I survey some of the connections between formal languages and number theory. Topics discussed include applications of representation in base k, representation by sums of Fibonacci numbers, automatic sequences, transcendence in finite characteristic, automatic real numbers, fixed points of homomorp ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
. I survey some of the connections between formal languages and number theory. Topics discussed include applications of representation in base k, representation by sums of Fibonacci numbers, automatic sequences, transcendence in finite characteristic, automatic real numbers, fixed points of homomorphisms, automaticity, and kregular sequences. Key words. finite automata, automatic sequences, transcendence, automaticity AMS(MOS) subject classifications. Primary 11B85, Secondary 11A63 11A55 11J81 1. Introduction. In this paper, I survey some interesting connections between number theory and the theory of formal languages. This is a very large and rapidly growing area, and I focus on a few areas that interest me, rather than attempting to be comprehensive. (An earlier survey of this area, written in French, is [1].) I also give a number of open questions. Number theory deals with the properties of integers, and formal language theory deals with the properties of strings. At the interse...