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12
Average State Complexity of Operations on Unary Automata
, 1999
"... . Define the complexity of a rational language as the number of states of its minimal automaton. Let A (respectively A 0 ) be an n-state (resp. n 0 -state) deterministic and connected unary automaton. Our main results can be summarized as follows: 1. The probability that A is minimal tends t ..."
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Cited by 23 (3 self)
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. Define the complexity of a rational language as the number of states of its minimal automaton. Let A (respectively A 0 ) be an n-state (resp. n 0 -state) deterministic and connected unary automaton. Our main results can be summarized as follows: 1. The probability that A is minimal tends toward 1=2 when n tends toward infinity, 2. The average complexity of L(A) is equivalent to n, 3. The average complexity of L(A) " L(A 0 ) is equivalent to 3i(3) 2 2 nn 0 , where i is the Riemann "zeta"-function. 4. The average complexity of L(A) is bounded by a constant, 5. If n n 0 P (n), for some polynomial P , the average complexity of L(A)L(A 0 ) is bounded by a constant (depending on P ). Remark that results 3, 4 and 5 differ perceptibly from the corresponding worst case complexities, which are nn 0 for intersection, (n \Gamma 1) 2 + 1 for star and nn 0 for concatenation product. 1 Introduction This paper addresses a rather natural problem: find the averag...
Transcendence of power series for some number theoretic functions
"... We give a new proof of Fatou’s theorem: if an algebraic function has a power series expansion with bounded integer coefficients, then it must be a rational function. This result is applied to show that for any non–trivial completely multiplicative function from N to {−1, 1}, the series P∞ n=1 f(n) ..."
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Cited by 5 (4 self)
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We give a new proof of Fatou’s theorem: if an algebraic function has a power series expansion with bounded integer coefficients, then it must be a rational function. This result is applied to show that for any non–trivial completely multiplicative function from N to {−1, 1}, the series P∞ n=1 f(n)zn is transcendental over Z[z]; in particular, P∞ n=1 λ(n)zn is transcendental, where λ is Liouville’s function. The transcendence of P∞ n=1 µ(n)zn is also proved.
(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.
TRANSCENDENCE OF GENERATING FUNCTIONS WHOSE COEFFICIENTS ARE MULTIPLICATIVE
, 2009
"... Let K be a field of characteristic 0, f: N → K be a multiplicative function, and F(z) = P n≥1 f(n)zn ∈ K[[z]] be algebraic over K(z). Then either there is a natural number k and a periodic multiplicative function χ(n) such that f(n) = nkχ(n) for all n, or f(n) is eventually zero. In particular, ..."
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Cited by 3 (1 self)
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Let K be a field of characteristic 0, f: N → K be a multiplicative function, and F(z) = P n≥1 f(n)zn ∈ K[[z]] be algebraic over K(z). Then either there is a natural number k and a periodic multiplicative function χ(n) such that f(n) = nkχ(n) for all n, or f(n) is eventually zero. In particular, the generating function of a multiplicative function f: N → K is either transcendental or rational.
TRANSCENDENCE OF THE GAUSSIAN LIOUVILLE NUMBER AND RELATIVES
"... Abstract. The Liouville number, denoted l, is defined by ..."
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Abstract. The Liouville number, denoted l, is defined by
Thue, combinatorics on words, and conjectures inspired by . . .
- JOURNAL DE THÉORIE DES NOMBRES
, 2014
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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.
