### 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 ξ_{N} → ∞, the maximal number of disjoint ε-pseudo-Behrend sequences included in [1, N] is (log N)^{log 2}e^{O(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 + ε_{N}), (a, b) = 1 and ε_{N} := (log N)^{1-log 3}e^{ξN√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, N^{1+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 2}2^{-zN√log2 N} with z_{N} → ∈ 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 language | French |
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Pages (from-to) | 181-203 |

Number of pages | 23 |

Journal | Discrete Mathematics |

Volume | 200 |

Issue number | 1-3 |

DOIs | |

Publication status | Published - 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

*Discrete Mathematics*,

*200*(1-3), 181-203. https://doi.org/10.1016/S0012-365X(98)00327-6