Ensembles de multiples de suites finies

P. Erdos, G. Tenenbaum

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

A Behrend sequence is a (necessarily infinite) integer sequence script A sign with elements exceeding 1 and whose set of multiples script M sign(script A sign) has logarithmic density μ(script A sign) = 1. By a famous theorem of Davenport and Erdos, this implies that script M sign(script A sign) also has natural density equal to 1. An ε-pseudo-Behrend sequence is a finite sequence of integers exceeding 1 with μ(script A sign) > 1 - ε. We show that, for any given ε ∈]0, 1[ and any function ξN → ∞, the maximal number of disjoint ε-pseudo-Behrend sequences included in [1, N] is (log N)log 2eO(ξN√log2 N). We also prove that, for any given positive real number α, there is a positive constant c = c(α) such that c < μ(script A sign N) < 1 - c where script A sign N = script A sign N(α) is the set of all products ab with N <a≤N1+α, a<b≤a(1 + εN), (a, b) = 1 and εN := (log N)1-log 3eξN√log2 N. This provides, in a strong quantitative form, a finite analogue of the Maier-Tenenbaum theorem confirming Erdos' conjecture on the propinquity of divisors. A similar result holds for the natural density of the set of all integers n such that F(n) has a divisor in the interval ]N, N1+α], where F is any polynomial with integer coefficients, and we establish in full generality that this quantity tends to a limit as N approaches infinity. Finally, we show that, for large N and q = (log N)log 22-zN√log2 N with zN → ∈ ℝ, the divisors d of an integer n with d≤N avoid no invertible residue class mod q with probability approximately Φ(z), where Φ denotes the distribution function of the Gaussian law.

Original languageFrench
Pages (from-to)181-203
Number of pages23
JournalDiscrete Mathematics
Volume200
Issue number1-3
DOIs
Publication statusPublished - Apr 6 1999

Keywords

  • Distribution of arithmetic functions
  • Distribution of divisors
  • Distribution of prime factors
  • Erdos conjecture
  • Multiplicative properties of polynomial values
  • Sieve

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

Cite this